Can I take the borrower's statement without reference to any available documentation that this is a refinance under HMDA and report it on our LAR?
To answer this question see 4(a)(3)Purpose on page Page D-10 of the GIR.
FWIW we would report this loan as a purchase. The son is purchasing full ownership in the property. Others may argue he already has ownership, and he does have partial ownership. But without "purchasing" the other 2/3 ownership from his parents he can not do anything with that property without their consent nor can they do anything without his.
BTW, you didn't say, but if the property is currently the son's primary residence Reg. Z's ROR will apply.
If you are unsure on which way to report or not to report, and based on your past problems, then pick up the phone and ask your examiner for guidance.