  In this paper, we will establish many explicit relations between parametric
Ap\'{e}ry-type series involving one or two parametric binomial coefficients and
Hurwitz-type multiple zeta values (with $r$-variables) by using the method of
iterated integral. Furthermore, we also establish some new identities of
integrals involving multiple polylogarithm functions and Kaneko--Tsumura {\rm
A}-functions in terms of Hurwitz-type multiple zeta values. Then, using these
formulas obtained, we obtain the explicit formulas of Arakawa-Kaneko zeta
values and Kaneko-Tsumura $\psi$-values. Moreover, we can find a symmetric
formula of Kaneko-Tsumura's conjecture about double $T$-values. Finally, we
define and discuss the Hurwitz-type multiple zeta values of parametric binomial
coefficient.
