Must I report a foreign account shared with my parent if the money is theirs?

The FBAR regulation gives a financial interest to each U.S. person in whose name an account is maintained. Who supplied the money doesn't matter. If you aren't an owner but hold a power of attorney that lets you move money from the account, that can be signature authority, which is also reportable.⁠^{[1]Source 1}⁠^{[2]Source 2}

Your parent holds $40,000 abroad and adds you as a named joint owner to simplify estate planning. You are a U.S. citizen. You have never deposited or withdrawn money. You must file an FBAR and report the account at $40,000.⁠^{[1]Source 1}⁠^{[3]Source 3}

Check how the bank has registered you: as a joint owner, under a power of attorney, or with view-only access. As a joint owner, report the full value in Part III. With a power of attorney that lets you move money, report the account as signature authority.⁠^{[2]Source 2}⁠^{[3]Source 3}