I always cringe at these threads because it's only a matter of time before someone will wheel out the old "people who are childless cannot understand." And it's not true, not at all. However, it is true that it is difficult to pick up the insight any other way than actually caring for kids - people never seem to talk about it. IME, nobody really wants to listen to it.
So, for my own sanity, some answers to a couple of your comments:
I do think parents are busy. However some parents do seem to be able to be social others aren't.
But the workload with kids varies massively depending on the ages - both in terms of the kids' behaviour and also whether they are in nursery/school, the age gap (close in age kids tend to be harder in the early years, easier as they get older), the number of kids, whether there are any family/friends who can do babysitting, whether there is any money for private nursery or childminder or babysitter, whether they nap or sleep well - and that's to say nothing of illness or special needs. So you know I'm not surprised if there's a difference in your parent friends' sociability.
I do think I'm busy myself too though - I have a FT job which demands unpaid overtime ie I never just work 9 till 5. I also spend at least another two evenings a week on a volunteering activity for my local community.
That is busy, but when my kids were under 2 my working day with them lasted from 6am (when I started cooking lunch/dinner for them, as I couldn't do it when they were awake - not enough uninterrupted time) until 7.30pm, with about half an hour for lunch while they were napping. During the time they were awake I would need to intervene with one or other of them - pay active attention, pull them out of harm's way etc - I would say about once every 2 minutes average. It was not possible to carry on a conversation - it was hard sometimes even to be able to, e.g. use a supermarket checkout, to say the things I needed to to be able to pay etc, there was so little respite from the two of them. Weekends were better as my DH was there but that just took it from being horrific to doable. Sometimes we would offer each other a solo day, one of us look after the kids solo while the other gets a day off - but the tiredness of having to do 6 days a week solo to "earn" a day off in return was too hard and we gave it up.
I was lucky in that my kids slept fairly well - that means, for the first four months whoever looked after them in the night never got more than 1h 20 uninterrupted sleep. DH did the weekend shifts and also a Thursday night, took Fridays off as holiday, but after three months his work started complaining about this arrangement so I did an additional month of hardly any sleep, 5 days a week. (And I had it easy, as I didn't breastfeed and could share the night feeds). After that things got better: the feeds fell into a routine and spaced out to every 4 hours, so we just had to feed them at 11pm and 3am. DD2 kept the 3am feed until she was 13 months old despite us trying very hard to wean her off it. Sleep goes out the window when they are ill of course, and what tends to happen is that even with something as minor as a cold you have 10 days' crap sleep as one and then the other gets ill, and then after that you get ill, but now you're also massively sleep deprived - and overworked, if you're working, as you'll have had (unpaid) time off to look after the kids while they were ill and the work will be mounting up on your desk.
So yeah, I guess there are childless people who're busier than that - but I suspect those are like on-call junior doctors maybe, and of course other carers, particularly elderly carers of elderly partners. I think those people are busier than parents of small children. I don't think many other people are as busy as that.
(Having said that, my kids are 4 now and I have all kinds of free time even while I am looking after them. So if you hang on a year or two it will probably sort itself out with your friend)