In this paper, we consider various combinatorial aspects of a family of polynomials, denoted by L_n^(α,β) (x), whose coefficients S_α,β(n,k) correspond to a special case of the partial r-Bell polynomials. Among the particular cases of L_n^(α,β)(x) are the generalized Laguerre polynomials, associated Lah polynomials, and polynomials arising in the study of hyperbolic partial differential equations. Here we provide a combinatorial treatment of L_n^(α,β)(x) and its coefficients, which were studied previously strictly from an algebraic standpoint. In addition to providing combinatorial proofs of some prior identities, we derive several new relations using the combinatorial interpretations for L_n^(α,β) (x) and S_α,β(n,k). Our proofs make frequent use of sign-changing involutions on various weighted structures. Finally, we introduce a bivariate polynomial generalization arising as a distribution for a pair of statistics and establish some of its basic properties.