The "old guy" you're likely referring to in Fossil Fighters: Champions is Professor Diggs, the museum curator.

He's not actually in the Parchment Desert itself, but rather in the museum. He's the one who gives you the initial instructions on how to find fossils and how to battle with your Vivosaurs.

However, he does appear in the "Fossil Frenzy" mode, which takes place in the Parchment Desert.