In this paper, by using special integrals and integer powers of the $\arcsin$ function, we obtained the recurrences or explicit expressions of some parametric Euler-type series involving multiple harmonic sums and multiple $t$-harmonic sums. According to our results, these Euler-type series are all expressible in terms of $\pi$, $\ln (2)$, and zeta values. In particular, by specifying the parameters, we presented as examples the evaluations of some special series, including some known ones in the literature and some new ones.