In this paper, we study infinite series involving $r$-Stirling numbers of the first kind and harmonic numbers. As a result, we establish the integral expressions for three parametric $r$-Stirling series, and show that these series are ultimately reducible to alternating multiple zeta values (AMZVs). As applications, the explicit AMZV expressions for some parametric series are determined, including some results presented by Wang and Chen, and some results similar to those of Hoffman.