Congratulations on your marriage!
My opinion is that since you already have a child, you are way too "out of the nest" to be projecting the impression that your parents are marrying you off. You are already an independent person arranging your own wedding, so your invitation should reflect that. Leave your parents' names off entirely. Just say, "Betty Boop and Christopher Plummer invite you to witness their exchange of wedding vows, Saturday july 25th, blah blah blah at blah blah country club, blah blah. You see?
Who pays, again, I don't know your age, but I assume you are an adult. Neither parent is obligated. You are an independent, fully functioning adult and should pay for your own wants and needs. (Traditions don't apply, since you have not lived traditionally, waiting until AFTER marriage to start a family. Since you went ahead and did things your own way, it's unfair to appeal to tradition NOW.)
However, you can certainly ask all the parents involved, "Peter and I are fully prepared to pay for our own wedding, so please don't feel obligated. We know that many parents WANT to contribute to their kids' weddings, so we thought we'd ask if you would LIKE to make a contribution in any way. Again, don't feel obligated, but we would love to have you participate."
You could simply say at the bottom of your invitation: reception with light appetizers to follow. Formal attire, please.
As it happens, I myself did our names only and "would like to contribute, don't feel obligated" when it came to my wedding. We had a modest but tasteful (I hope!) wedding. All these details are great ways to demonstrate the independence and responsibility that are necessary for maintaining a healthy, stable marriage. 16 years and counting!
best wishes.