The paper focuses on the solvability and computability of the parameterized polynomial inverse eigenvalue problem (PPIEP). Employing multiparameter eigenvalue problems, we establish a sufficient solvability condition for the PPIEP. Three numerical methods are used to solve PPIEPs. The first one is the Newton method based on locally smooth $QR$-decomposition with the column pivoting and the second the Newton method based on the smallest singular value. In order to reduce the computational cost of computing the smallest singular values and the corresponding unit left and right singular vectors in each iteration, we approximate these values by using one-step inverse iterations. Subsequently, we introduce another method — viz. a Newton-like method based on the smallest singular value. Each of three methods exhibits locally quadratic convergence under appropriate conditions. Numerical examples demonstrate the effectiveness of the methods proposed.