The question is whether or not my parents will agree.
Literal
Problem [topic-は] parents [subj-が] agree-[favor-てくれる][whether-かどうか] [copula-です].
~かどうか ('whether or not X') is the standard way to embed a yes/no question as a clause. 問題は~かどうかです ('the question is whether or not...') is a very common setup in formal discussion. 同意する takes its object with に.