I’m pretty sure you can disambiguate this by saying “Ils sont restés avec ses parents à lui” or “Ils sont restés avec ses parents à elle,” using the “possessive à“.

The French.about.com page “French Possessive Adjectives,” by Laura K. Lawless, mentions this construction, but seems to indicate that the technically correct way of punctuating it would be “Ils sont restés avec ses parents, à lui” or “Ils sont restés avec ses parents, à elle.” I wouldn’t pronounce a pause here, but often these kind of “double pronoun” constructions in French are supposed to be written with a comma, regardless of the pronunciation, because the formal interpretation of them is as parenthetical clauses adding more information.

Also, you cannot replace “lui” with “elle” in

Il lui a souhaité bonne chance.

In this context (immediately preceding a verb), lui is the only possible form of the feminine indirect object pronoun. Before a verb, “elle” is only used as a subject pronoun. Elle is used as an object pronoun after a preposition. The following question mentions the stressed indirect object pronouns: Quels pronoms (toniques ou normaux) peut-on utiliser pour un COI introduit par « à » ?

Apparently, you should use “à elle.” Laure mentions in a comment that the way of saying it would be

Il lui a souhaité bonne chance à elle.

with both “lui” and “à elle,” but this would only be used when contrasting “à elle” with some other possibility such as “à lui.” I don’t know if a comma before “à elle” would be possible or preferred in this sentence.

First, Il elle a souhaité bonne chance is not correct. You must use lui in this case, no matter the gender.

Second, there are too possibilities. You can add “à elle“, it’s not super correct but orally it’s fine, if it’s necessary to disambiguate. But you can break it down like this :

Either there is no possible confusion and you don’t need disambiguation, if there is one set of parents, or if there are two or more set of parents, you can always say “Ils sont restés avec les parents de Marie“.

And if you don’t know the name of anyone there, I’m pretty sure saying “à elle” is fine in any context.

The sentence

Ils sont restés avec ses parents.

Carries the information that Some people, designated here by “Ils” stayed with the parents of some person, designated here through “ses”. For the sentence to be understood you have to give the informations about these persons in some other sentence, for example:

-Sais-tu où sont Bob et Alice ? -A Toulouse. Clara leur a proposé de passer un peu de temps dans le Sud alors ils sont restés avec ses parents.

Here “Ils” refers to Alice and Bob and “ses” to Clara. No ambiguity.
If you think “ses” is ambiguous in who it refers to, replace it with the name of the person:

Ils sont restés avec les parents de Clara.

Or with any way to designate her/him:

Ils sont restés avec les parents de la boulangère.

The same goes for your second sentence:

Barbara avait une épreuve à passer. Alfred lui a souhaité bonne chance.

You can say:

Alfred a souhaité bonne chance à Barbara.