to not mind the term 'babysitting'

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I know loads of people really dislike the term "babysitting" when applied to a parent in sole care of their child whilst the other parent is doing a specific activity. But I don't mind it. I use it loads with DH: "I'll babysit, you pop to the shops". Or "are you ok to babysit if go to Xs for drinks?" etc. I use it in reference to both what I am doing e.g. "I can't, I have to babysit as DH is at X" and to DH "Oh yes, night off, DH is babysitting".

I think it's a concise to way to say, I will be in charge of the kids whilst you go out or you do something where you cannot look after them. Another way of saying "I'm looking after the kids" or "I'm parenting alone".

I think it is different to (or at least a specific part of) "parenting" which is something I do whether the children are in my presence or not and can be done jointly or solely.

Yet I have some friends who take real issue with the term. AIBU to use it?

Not unreasonable if you are comfortable with it but not a, term I'd use. More likely to say are you alright with the kids tonight.
I think a lot of people resent it as, it's often said to men. The idea is that looking after your own children is parenting not babysitting.

It's OK if it used for you both I think, I've used it it before but more in a kind of a jokey way, eg :

Friend : Do you want to call over tonight for a few drinks?
Me : Nah, I can't tonight I'm babysitting as DH is out

I think if it's only used when talking about the father and the mother is the only one who has to book her time off and arrange him to babysit, then it's a problem

I think it all depends on how it’s used, context etc.

Sadly in my circle it's not used in that way.
It's a 'I hope DH will babysit so I can come out for tapas' or partners saying 'I can't do any other menial task, I'm babysitting'

Babysitting means "to take temporary care of a child or children in the absence of a parent or guardian"

So I think it's a bit odd to use the phrase when you are talking about your own children.

It is never babysitting when they're your own DC. It's fine to use it, it's fine to be fine with it, but it's still not babysitting.

I've had the odd person ask me if DH is babysitting when I've been out. It isn't important enough to bother me, but it's not something I'd use.

Odd phrase as a parent- either male or female. If I’m invited out and my husband has plans already I’ll just say I can’t come husband is out that night. It’s a given you are with your kids as a parent, to say babysitting implies it’s a given that you aren’t with them unless you are needed to be.

When it is only used in the context of a husband or male partner then I think it is completely wrong. A babysitter is someone from outside of the household.

In your case as both of you say it that is acceptable but in general the above is what is what is usually being referred to.