We show that a linear partial differential operator with constant coefficients P ( D ) is surjective on the space of E-valued (ultra-)distributions over an arbitrary convex set if E ′ is a nuclear Fréchet space with property (DN). In particular, this holds if E is isomorphic to the space of tempered distributions S ′ or to the space of germs of holomorphic functions over a one-point set H ( { 0 } ) . This result has an interpretation in terms of solving the scalar equation P ( D ) u = f such that the solution u depends on parameter whenever the right-hand side f also depends on the parameter in the same way. A suitable analogue for surjective convolution operators over R d is obtained as well. To get the above results we develop a splitting theory for short exact sequences of the form 0 ⟶ X ⟶ Y ⟶ Z ⟶ 0 , where Z is a Fréchet Schwartz space and X, Y are PLS-spaces, like the spaces of distributions or real analytic functions or their subspaces. In particular, an extension of the ( DN ) - ( Ω ) splitting theorem of Vogt and Wagner is obtained.