We study the approximation of the distribution of X_T, where (X_t)_{t ∈ [0, T]} is a Hilbert space valued stochastic process that solves a linear parabolic stochastic partial differential equation driven by an impulsive space time noise, dX_t+AX_tdt = Q^1/2dZ_t, X₀=x₀ ∈ H, t ∈ [0,T]. Here (Z_t)_{t∈[0, T]} is an impulsive cylindrical process and the operator Q describes the spatial covariance structure of the noise; we assume that A^{- α} has finite trace for some α > 0 and that A^βQ is bounded for some β ∈ (α - 1, α]. A discretized solution (X_hⁿ)_{n∈{0,1,...,N}}} is defined via the finite element method in space (parameter h > 0) and a θ-method in time (parameter Δt = T/N). For ∈ C²_b(H;ℝ) we show an integral representation for the error {pipe}Eφ(X^N_h)-Eφ(X_T){pipe} and prove that, where γ < 1 - α + β. This is the same order of convergence as in the case of a Gaussian space time noise, which has been obtained in a paper by Debussche and Printems (Math Comput 78:845-863, 2009). Our result also holds for a combination of impulsive and Gaussian space time noise.