Also to be honest if your child is continuously excluded you need to ask yourself why and not blame other parents.
There is a girl in my daughter's class who is a bully and making my daughter's life a misery to the point where she doesn't want to go to school some days. She constantly secretly pinches, scratches, kicks my daughter under the table etc, also she name calls her. As far as I know she does it to the majority of girls, to whoever is not feisty and she feels she could get away with. She only has one semi-friend, and I wouldn't be surprised if she wasn't invited to birthday parties.
I had to tell the teacher a few times, but I'd never approach the parents, and to be honest I have no idea if they are at all aware that their daughter is a bully, they could well be wondering why she isn't popular...
I'm not blaming the child btw, she obviously has issues and needs help, but I'm not throwing my child under the bus and ever inviting this girl to any parties or playdates.
Another thing could be that your child is just very introverted, quiet, special needs, etc, shy, doesn't make friends easily etc. Then don't get dramatic on their behalf, and create a plan where you expand their circles and meet as many people as possible at other activities similar to the ones that your child enjoys. There would be a similar child or two out there somewhere, they don't have to have friends only at school. Though even at school if I was a mother of such a child I'd probably would try one on one playdates with every child in the class that my child would want to have round, and see if they could build a friendship outside of a classroom initially.
And have things in their life for them to look forward to, have stuff to do at weekends even if it's flying a kite in the park or getting a hot chocolate somewhere, don't rely on other parents and kids to organise a social life for your child and then seethe that they don't! Nobody owes you and your children anything.