Does HOEPA apply to a loan where the collateral for the loan is the borrower's parent's primary residence which is, also, the borrower's primary residence? The borrower is not the property owner and the parent's signatures will not be on the note.

Section 32 applies to transaction secured by the consumer's principal residence. It does not say "the borrower's" principal residence.

The loan you describe IS secured by a consumer's principal residence....therefore it would be covered. Part of the purpose of this section of Reg Z is to guard against predatory lending for loans secured by homestead property to prevent consumers from losing their homesteads.