"Accompany" includes the "with", so "accompany" is identical to "go along with" (or "go with").
In AE, "go with" is neutral - typically it means all parties are equally involved. "Go along with" is common for things like trips, and implies the verb subject ("parents") is less important. For example, if "Chinese parents go along with their children" on a trip, then the trip is about the children, and the parents are just "going along". The reason for the parents going along is not implied.
"Accompany" can be neutral (no implied reason), or it can imply that the children are too young to travel alone.
A common AE saying that uses "along" is "along for the ride". If 3 friends are driving to a store to do shopping, their friend Sam (who does not plan to shop) may "come along for the ride": he will drive there and back with them, just to enjoy the trip.