The solutions of your equation can be injected into the solutions of the $S$-unit equation over $\mathbb{Q}$, where $S=\{\infty,2,3,5,p\}$. As the latter is known to have finitely many solutions by the results of Siegel, Mahler, Lang (see Ch. 5 in Bombieri-Lang: Heights in Diophantine geometry), your equation also has finitely many solutions.
By a result of Beukers and Schlickewei (see Theorem 5.2.1 in the above mentioned book), the number of solutions is at most $2^{40}$ for any given prime $p$.