We consider a semilinear parabolic stochastic integral equation (Formula presented.) Here t ∈ [0, T], ω in a probability space Ω,x In a σ-finite measure space B with (positive) measure Λ. The kernels aμ(t) are multiples of tμ-1. The operator A: D(A) ⊂ Lp(B) → Lp(B) is such that (–A)is a nonnegative operator. The convolution integrals aβ ⁎ Gk are stochastic convolutions with respect to independent scalar Wiener processes Wk. F: [0, T] × Ω × D((-A)θ) → Lp(B) and G: [0, T]×Ω×D((-A)θ) → Lp(B, l2) are nonlinear with suitable Lipschitz conditions. We establish an Lp-theory for this equation, including existence and uniqueness of solutions, and regularity results in terms of fractional powers of (-A) and fractional derivatives in time.