Good replies, especially from Manda -- take a deeper look at your own self-esteem in the larger picture.
Also: Kids' sporting events and school events are just not always good places to try to build up acquaintanceships through chit-chat. Sometimes they are, but often the parents have known each other for years already, their kids see each other outside the sport or are in the same class at school or they all go to the same church, ....So they tend to turn to the person whom they know already and probably talked to yesterday as well.
Not polite, I agree, but you may indeed have to introduce yourself to that new person who walks up to the group; however, you do need to stop wanting their attention so much.
Do they have any input into your child's playing position in a sport, or your child's role in a school event? Nope. You seem to be worried that if you aren't talking with some specific in-group of moms, your child is "missing out on opportunities" -- what opportunities do you mean? What do you think these parents and their kids can provide for your child if you get their attention? If you like these parents and want to know them for their own sakes -- and I've made some good friendships through my kid's activities with adults I like for themselves -- it's great to work on friendships. But it sounds as if you want to pursue conversations because you feel these parents are social gatekeepers to other things, events that you know happened but weren't invited to. Do you really want to be part of the crowd if you feel this uncomfortable trying to get in the door?
Of course be cordial and say hello to them; you know who they are (which is not the same as "you know them", nor do they really know you). Then when the mom you're talking with turns to someone else, just excuse yourself, leave, watch your kid play the sport or occupy yourself by helping out with the event. Maybe be the one who goes up to some other mom who's there on her own and introduce yourself and be the one to include HER.
Are you volunteering to take on tasks for your child's activities? That can really help you make some friendships that are much more real that just chatting during an event or game. And it puts you (and your child) on the radar of the coaches and teachers and PTA and generally helps everyone get things done. You'll be too busy volunteering during games or class events to worry about whether the other parents will include you in conversation, and you'll be benefiting your child, the team or class, and yourself.
You mention that you don't have as much money or status as others do. Please explore why you feel that matters so much. I really doubt these moms are intentionally trying to exclude you or make you feel "not good enough." Whenever I've felt like someone was doing that to me, I learned to remind myself that that person actually hasn't expended a single thought on me and isn't worrying about my worth - so why should I expend any mental energy on him or her? You need to be able to do that, to stop caring about getting the attention of people whom you perceive as somehow important ("I don't want my children to miss out....") but who are only as important as you let them be inside your own mind.