Is this an adjective phrase? Is 'when her husband mentions an invitation to a dinner party' an adjective phrase? I think 'to a dinner party' is...

Your first phrase -- "'when her husband mentions an
invitation to a dinner party" -- is not an adjective phrase in its function. In its
function, it's an adverb phrase. Adjective phrases (whether just one word or a long
series of words) modify nouns or pronouns and answer questions such as "which?" or "what
kind of....?" Adverb phrases modify verbs (and a lot of other things, just not nouns)
and answer questions such as "when?", "where?", "how?", "why?", or "to what extent?"
Some grammarians classify adverb phrases based on which questions they answer, giving us
adverbs of time or condition ("when?") or place or manner or reason or
extent.

You're right about the second phrase -- "to a
dinner party." It's a prepositional phrase and modifies
"invitation."

Prepositional phrases often work (or
function, as you might hear it called) as adverbs. "They went to a dinner party" offers
an example. In this sentence, "to a dinner party" modifies "went," tells where, and thus
is a prepositional phrase functioning as an adverb.

As a
rule, of course, you need to look at the entire sentence to be able to determine if
something is an adverb or adjective.  Definitions of (and terms for) parts of speech and
grammatical functions also vary a little from source to source; you need to look
carefully at the defintions and terms that you're working with.