OP, unfortunately you've expressed much of your problem in terms of your parents being a service provider; wipes, toys, babysitting. They do babysit for you, though the lack keenness could be addressed. Just say, you don't seem terribly keen, is there a problem. Unfortunately the answer may turn out to be they didn't want to do it but had agreed with rather thin lips. I hope not.
You say they pop over, so they are visiting you. Do you want something different, stay for a meal, you don't make clear so it's hard to comment.
If you had such a close relationship with your mum, then surely you can approach her about this (not the service providing, but the many chats)
As for apparent unreasonableness of settling the new dog, then they're NBU; having two young children about - you don't say for how long - could well be unsettling for the dog. If it's a puppy, then your parents be constantly on the alert for piddling and not up for entertaining of any kind. Having done this myself, I can vouch for how time-consuming it is. I was fit for no-one by the end of the fortnight's house-training.
There seem to be two issues; not being as close to your mum - by the way you can up the number of phone calls yourself. Is it always you who calls?
The issue of now being "useful" is quite another, and one you may have to let it go.
Your MIL, though. Yikes, that's just nasty.