To not take child to party?

Exactly this. Decline. It’s no biggie.

I have 3.5 year old boy girl twins.

Again, I have never seen anyone dropping kids that young and leaving.

It's terribly rude and not expected around here, until at least year 5 anyway,

I wouldn't have had a clue! We all picked up at different times, so I would only have a tiny snapshot of a few parents. I was also picking up 2 for a couple of years, different ages, so even less aware of what else was happening.

Where would you see a list of names, sounds a bit worrying that you would see something with full names of children!?

OP, someone upthread said do what's easiest. I second that advice.

Where is the party being held? If it's a softplay center could you pay for your other child to enter?

I would just decline and take your twins on a day out. Presumably if this child is turning 5 they're about to leave nursery and won't be inviting your DC next year anyway so not a lasting friendship. Your twins have years of soft play parties ahead of them, this is hardly an "opportunity" . It's much easier for your twins to be separate people once they start school, I wouldn't be starting with the whole 'resilience' crap just yet.

Depends on the nursery. My kids went to a private nursery and you could collect any time. I think we paid til 5 but unless I had a work meeting I would pick up around 4.

When mine then attended a school nursery you all had to wait in a line to collect them. If you were at the front of the line you wouldn't necessarily see who is collecting at the back. If either parent has other children in the school they might collect them first before going to nursery or vice versa, so some parents might never see who picks up who.

Nurseries/schools should never give out children's details to other parents with their surnames on, it would be a GDPR/safeguarding breach. First names only.

It's very feasible the other mother has never met OP or her children. At private nurseries you pick up and drop off at a time which suits you (between opening and closing time). Parents don't all wait outside together for the same 'pick up time' like in school nurseries/pre-schools. I assume OP's children go to a private nursery, as most term time settings have finished now, but I could be wrong.

OP's child is 3.5
That's young to be left alone.

They can breathe separately.

I'd do this as it's probably a mistake. Just explain it's a childcare issue. If I was the mum running the party I wouldn't mind at all and probably think "oh shit I only invited one twin!"

Not at nursery. School maybe.

Someone in DD’s friendship group does this, without asking. Their DD sibling then has a breakdown and spoils the party.
I don’t mind a sibling coming if it’s a public jump party for example as paying customers can walk in.
However with this child you have to prepare a party bag for it, serve her first or she’s goes absolutely nuclear, (there is no extra needs) she’s just a brat, mum practices gentle parenting and the elder sibling has to hand over her party bag if someone doesn’t have enough. The mother is extremely entitled and brings the sibling even to private parties in your own home and the mother also stays. Doesn’t even ask if it’s ok, it’s like she wants them to do everything together, all the time.
The elder sister is fighting for her freedom 😆

It’s A 5yo party. IME they are drop and run leave!

Do you really believe that the nursery mum hasn’t noticed the twins being picked up by one parent. Or the list from the school that has matching surnames that isn’t a guarantee they are related but given you see one person picking up 2 kids it isn’t much of a leap.

I'm all for encouraging twins to have separate friendships and agree they need to learn that sometimes one will be invited to do things the other one isn't, and that's ok. However, I also think 3.5 is too young for them to understand this. Once they're at school, it will be easier. Especially if you choose a school with 2 form entry and ask for them to be in different classes to encourage individuality and different friendships. It's so much easier to explain at 5, as they're so much more mature by then and beginning to understand more that they are individuals and can be friends with different people. At 3, I would politely decline.

Likelihood is, the other child's mum has just asked him to list friends he plays with and wishes to invite and is oblivious that your daughter is a twin. I'm guessing this child is leaving for school imminently (early September birthday, or August deferred entry from last year?) so it's unlikely the friendship will continue beyond the next few weeks. I'd decline and think nothing more of it. As I said, once they're in school I think it's perfectly acceptable to explain this concept to them, but 3 1/2 is still extremely young.

Because it's unlikely the mum knows they are twins.
The party mum will have either asked the nursery for names of children her son plays with and may have even said can you choose the 8 friends he plays with the most. Or as he is turning 5 he is likely to have told his mum who he wants at his party. Again she might have said can you tell me 8 friends you would like to come. He's hardly likely to say oh there are twins but I have only named one. He's likely to only name the children he likes.

I don’t understand why the parents of the 5 year old would only invite one of the twins…at this age why wouldn’t they both be invited. Weird

Sorry but I wouldn’t bother personally - the 5 yo whose party this is, I assume, is likely going to go to a new school in Sept and there is a full 1 or 2 academic year difference between them. They won’t be friends in any long term sense - nor are they now in any meaningful way given yours are only 3.5yo.

Just thank them but say you can’t make it and let it go. When they are older, in infant school themselves, and this happens again but with ‘real’ friends they’ve made they will be old enough to have it explained why one of them is not invited to a particular party.

Just not necessary to have to deal with this now, at this age, with this particular friend.

Most soft plays aren't exclusive booking. I'd message and say you're going to have to bring your son too and will pay for him to come in and distract when they go to the party room. Take a cupcake and a bag of tat for him for after and he probably won't even notice.
People needing to bring a sibling along is completely normal throughout infant school, the only time it's not ok is when you expect them to join the party and scarper leaving 2 kids instead of 1.

When a mum told me that for our soft play party I just added one on, it was super cheap per head once I'd paid the booking fee. There have been loads of siblings along at parties I've been to.

I mentioned your twins! This is an analogy: if you don’t give the twins a present on your daughter’s birthday - this is something for her and not them. Ditto party invitation.

I don't have twins but we have many sets in our family and I actually agree with you. Yes, they are individuals but I think twins stay a unit longer than most siblings and their egos develop slower. By the time they go to school they are normally fine apart.
Only you know what is best.

Sorry i was replying to someone else,.i should have quoted.

Does the parent know there is a twin? (Remember that even if they knew that at some point in the past, if they don't know you well then they may have forgotten that little Bella is one of the twins.) If they just asked the birthday child to make a list, they might just not have thought about the twin issue.

At this time of year, there may well be some declines as people are on holiday, so it's possible there might be a second round of invites. Maybe respond saying "I think dropping Bella off is going to be too difficult with Billy in tow and not invited. I think we'd better decline, but if you turn out to have some negative replies, and have space for both of them, they would be able to come. I haven't mentioned it to Bella at the moment."

It's different when they get a bit older, however; you don't want them to end up never being invited because people feel they can't invite just one. There are bound to be some parties which are "all the boys" or "all the girls", so that will help them get used to the idea that they won't always get the same invites.
(I remember someone being thrilled when my mother asked if it was okay for my brother to invite just one of her triplet sons for a playdate; they got very few invites because most people felt they would have to invite all three.)

Is it at all possible that the Mum doesn't know they are twins? You know your children best, but I think letting your daughter attend the party and taking your son to the soft play might work? I'm presuming the party children will be mixed with the general public for play, and just go off for food. If your son will understand that he is eating separately from your daughter, but still gets to play, it could work? You might need to have a little something for him as she will have a party bag though. I do think 3.5 is too young for them to understand the concept of them being individuals to much of an extent.