In this paper, the averaging principle is studied for a class of multiscale stochastic partial differential equations driven by $α$-stable process, where $α\in(1,2)$. Using the technique of Poisson equation, the orders of strong and weak convergence are given $1-1/α$ and $1-r$ for any $r\in (0,1)$ respectively. The main results extend Wiener noise considered by Bréhier in [6] and Ge et al. in [17] to $α$-stable process, and the finite dimensional case considered by Sun et al. in [39] to the infinite dimensional case.