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We consider a general class of $L^2$-valued stochastic processes that ariseprimarily as solutions of parabolic SPDEs on p.c.f. fractals. Using aKolmogorov-type continuity theorem, conditions are found under which theseprocesses admit versions which are function-valued and jointly continuous inspace and time, and the associated H\"older exponents are computed. We applythis theorem to the solutions of SPDEs in the theories of both daPrato--Zabczyk and Walsh. We conclude by discussing a version of the parabolicAnderson model on these fractals and demonstrate a weak form of intermittency.

Continuous random field solutions to parabolic SPDEs on p.c.f. fractals