Disambiguating ambiguous pronouns

Both examples sentences end up being being ambiguous because, as said, it's not clear who her refers to—Lily or her mom. It's also odd to refer to Lily and one parent—then refer to the other parent as the first parent's spouse.

The simplest sentence, that refers to both mother and father, is:

Lily's parents were preoccupied in conversation.

Or, to mention each specifically and in the order given:

Lily's mom and dad were preoccupied in conversation.

In both cases, move preoccupied in conversation to the end of the sentence, thereby not having to use the ambiguous pronoun her.

Note that you can avoid the ambiguity of the pronoun in the original sentence if it's the father who's mentioned first:

Lily's dad was preoccupied in conversation with his wife.

In this version, his can only refer to one person—Lily's dad. So, it's clear who everybody is.

While this avoids ambiguity, I don't personally feel it's as natural as either of the two previous examples—because it requires a bit more parsing. But that's just a matter of opinion.