I think what the Swiss require is standard, at least for Europe: birth certificates for both spouses + marriage certificate (with apostille to prove genuine).
I disagree, I think it is the rule (even in Jamaica) to list parents, unless the birth was out of marriage and the father didn’t agree to paternity. In any case, you seem to be paying omissions made by your parents. If your parents were american and temporarily living in Jamaica, it could be that your dad signed the USA documents but did not bother with the Jamaica ones for whatever reason. You could try to find from your parents what happened, but I suspect it is too late to amend. But as you say, they are just “asking” why the father is not listed, so all you have to do is provide the reason. PS: The Swiss generally try to be accurate and if they pick inconsistencies in official documents they will take their time to check (eg different father mentioned on USA and Jamaica documents). If you were american at birth, you should look at the USA consular report of birth this would list both parents.