I wasn't playing a lot of RPGs when my kids were in the first few instar stages, but I did play adult board games.
In general...
a) First instar humans basically eat, sleep, and poop and as such are easy to take care of in just about any situation. They don't have much patience though, so you have to take care of the immediately. This means that the adults need to supply the patience that the infants lack.
b) Second instar humans like all humans whose brains have gone through the initial compile stage need lots of intellectual stimulus, and in most cases also lots of social stimulus. The still also lack patience, so you have to take care of them immediately and usually continuously. They also become mobile and so require a lot more direct oversight as they have the manipulative digits to manipulate just about anything, but lack sufficient experience to attach causes to effects. Leave them alone and they'll break something, potentially themselves. This is probably the hardest period for getting your gaming in or really any other adult interaction. Your best bet here is to employ baby sitters. This can either be older children that enjoy playing with the little ones (third instars work), or it can be an 4th instar "preteen" or even a "teen" - though the later will usually expect some sort of payment.
c) Third instar humans are much like second instars, but they can reasonably left without direct supervision and usually actually prefer it most of the time provided they have a means of amusing themselves. This lets you use games, electronics, and even other children as sitters, and pretty much everyone will enjoy it as a change of pace.
d) Fourth and fifth instars will likely enjoy being incorporated in the adult gaming itself, and this can be a special pleasure for everyone involved. It may mean you have an especially large group or multiple tables though.
If your kids are beyond the eat, sleep, poop stage, but not yet big enough to be left in front of a movie or a pile of board games with the expectation of only needing minimal help from you for the next few hours, then I suggest you're probably at that point where a) you do actually need friends that know what this is like and are sympathetic and b) that the best adult games to enjoy during this period are the ones that don't require absolutely continual attention and involve frequent short breaks. It's also helpful if everyone is ok with the casual pace of play that is likely to dominate during this period. I suppose RPGs would be theoretically possible, but I think you'd run the risk of looking like a parent from 'The Guild'. Again, employ a baby sitter and split the costs if you plan this. Lavishing high amounts of attention on kids below age 5-6 is pretty much going to be expected. Also, I find that regularly abandoning them and leaving them with a single spouse isn't fun for anyone including the one that got away. It's hard to get enough quality time with 'littles' as it is and the more time they have being co-parented the more content they tend to be. The good news is that this is all pretty temporary and its pretty easy to go back to your pre-parent pastimes once they get a little older.
It's recently occurred to me that the biggest obstacle to families of gamers getting together to play is probably the size of the house required to handle all the gamers and their retinue. Given a group size of say seven, if a large percentage of the group isn't spouses of the other half, it would be pretty easy for the entire group to be over two dozen. Hopefully you have a player with 'Stately Wayne Manor' who is willing to host.