Just a thought, but if you did not let your kids go to their kid's birthday parties, don't start out inviting their kids to birthday parties for your kids or you are liable to get people thinking that you are just fishing for gifts for your kids, but didn't want to buy any for theirs. Start with other holiday events, then as you get to know people better, put the word out that you are now open to invitations for your kids to go to their kid's birthday parties because you have changed in your beliefs. Once you have that established, you can issue birthday party invitations.
Does that make sense to you? I just know how people are sometimes.