In this paper we study the initial boundary value problem for a class of integro-differential time fractional order parabolic equations with a small positive parameter ε. Using the Laplace transform, Mittag-Leffler operator family, C_{0}-semigroup, resolvent operator, and weighted function space, we get the existence of a mild solution. For suitable indices pin [1,+infty ) and sin (1,+infty ), we first prove that the mild solution of the approximating problem converges to that of the corresponding limit problem in L^{p}((0,T), L^{s}(Omega )) as varepsilon rightarrow 0^{+}. Then for the linear approximating problem with ε and the corresponding limit problem, we give the continuous dependence of the solutions. Finally, for a class of semilinear approximating problems and the corresponding limit problems with initial data in L^{s}(Omega ), we prove the local existence and uniqueness of the mild solution and then give the continuous dependence on the initial data.