General Obligations

I think there are some (rare) situations in which children may have an obligation to remain in contact with their parents even if they do not particularly wish to.  However, this is not due to any sort of familial connection, and can apply in any relationship.  Furthermore, the general principle can apply to any pairing of an action/gain nature.  I've tried to set it out below in as straightforward a manner as possible.

C = cost to oneself    G = the amount of good others gain from your action
In this setup, the minimum gap between C and G is +4.

1) A situation wherein a generally decent parent wants their child to stay in contact with them, and the child finds this slightly inconvenient (the child does not hold any friend-like feelings toward the parent)
C = 1    G = 8    The result is +7, so the child is morally obliged to stay in contact, unless C increases or G decreases.

2) A situation wherein a rather bad parent wants their child to stay in contact with them, and the child does not want to do so (their relationship with the parent is quite bad)
C = 5    G = 6    The result is +1, which is not enough.  If the child chooses to stay in contact with the parent, they will be acting in a supererogatory manner.

3) A situation in which a good parent wants their child to stay in contact with them, and the child shares that desire (they are friends)
C = -4    G = 8    The result is +12, so this is morally obligatory - however, the obligation is irrelevant, because the child wants to stay in contact with their parent anyway.

4) To illustrate that this does not only apply to parent/child relationships, or indeed any relationship at all, this example is slightly different.  In it, a person enjoys the taste of meat, but realises that the production of meat comes at a cost to farm animals.  Thus, the farm animals would gain from the person's becoming a vegetarian or vegan.
C = 7    G = 20   The result is +13, so this is morally obligatory.

This does not invalidate my earlier post about Dixon's confusion of sentiment with obligation.  After a friendship ends, unless one or both sides retain lingering sentiment, choosing to help a former friend over a stranger provides no more good to the friend than one could give to the stranger - the G numbers are equal.