OP, thank you for asking here (and also do not hesitate to ask the family who invited you).
Unless this family is close family friends with yours (sounds like no), the invitation is for your child only. The synagogue services are open to all, so you would be welcome to attend if you wanted to, but it wouldn’t be expected that you would stay with your child. The 12/13 year olds are typically dropped off for the service and will sit together. If you think there is a chance your son will have great difficulty sitting respectfully due to SNs or some other circumstance, it would not be a problem for you to stay and help him take breaks, etc., but there are synagogue volunteers who will be keeping an eye to make sure kid guests are supervised.
The party is not open to all, and the name that appears on the invitation is indicative of who is invited. They keep the kids super engaged at these things, it’ll likely be lots of fun. Feel free to drop off.
Thanks for supporting your kid’s friend.