Wedding invitation including both of the parents’ names

If you want to, go for it. It’s very “traditional”. I did not do this and my mom through a fit. But yes that would be the proper way of doing it. I think technically if one family is paying they’re supposed to be listed first?

"Together with their Parents" is usually used when the parents names aren't itemized. Another idea is to word it this way:

STEPHANIE

daughter of Mr. William and Mrs. Idy Chan

and
RYAN
son of Mr. Robert and Mrs. Fanella Sham

Invite you to their wedding

We put:

Together with their families,

Melinda

and

Jeffry

You definitely can! I've seen wedding invitations where it included the parents full names as well. For my invitation it just simply said "Together with their parents, Heather Coloma and Cody Robertson joyfully invite you..."