Both of his parents use social media. Neither appear to be involved in their son's fraud at all.
His father doesn't even post about Welsh history. His mother, who he claims is Native American, does not post any claims to this heritage, nor does this look to be an interest.
Lawrence Joseph Jones, Jr. / Tywysog Llywelyn Jones Cymru likely does have a paternal 3rd great grand father who came from Wales to Ohio to work in coal mines. This doesn't make him a Welsh citizen or give him a legit claim to royalty. This does not make him special.
His maternal heritage from Mexico is more recent. For instance a matenal great grandfather petitioned for naturalization to USA in 1923, he was born in Mier y Noriega, Nuevo León, Mexico. This great grandfather died in 1971, Texas.
Here is the obituary of his maternal grandfather
https://www.legacy.com/us/obituaries/sanantonio/name/frank-salazar-obituary?pid=88829362