When do you share the guest list with your parents?

Did you allow at least a couple of seats for each set of parents, for their closest friends? If you did, then you can say no to all additional people. If you didn't, then show them the list with 2 to 4 open spaces for your parents to fill, and 2-4 for his. Whatever you would compromise on. Unless a tiny under 20 type wedding.
You really need to show them the list. So they do not just assume all kinds of other people are on the list, and talk to them as though they are invited. Or, in case family, or friends, want a surprise type shower. ( 2 day notice to you if you are lucky!) Aunts or your friends may ask MOB or MOG who to invite. You do not want 10 people not invited, along with 10 who are at a shower. So giving them the final, no changes except for mistakes, guest list could save you from trouble down the line .

I'll have my mom and FMIL look over the list just to make sure there is not family that we may have missed. We both have large families and both of our families are paying for the wedding/reception, so I feel it's curiosity.

Okay, DH’s Mom provided a few names that she wanted to invite.
My 92 yo Granny had a few names, but once she reached 10, I told her that was all that she got.

I invited my Mom’s co-workers, 1st Cousins, and friends that I thought that she might have wanted to be there if she could’ve provided input.

Oh, the Paternal Unit, nor his relatives were invited. I did invite my 1/2 Sisters. They still hadn’t responded as of yesterday. My Wedding was 9/21.

😒

You don't have to add to your list, but sharing it will help your parents avoid awkward conversations with their friends if you've decided some of their friends won't be included.

My FH and I are keeping our wedding pretty small so we probably won't be checking with either side, maybe only asking about specific family members and family friends to make sure there's no drama on the big day.

My mom and his parents have had nothing to do with our wedding and I doubt they would care who we invite. So far the only thing his parents have done was give us a list of a couple relatives that my fiancé might've overlooked. We had asked for that though and we'll still have the final say on if we invite them