The result holds true in the parabolic case for F uniformly elliptic. For a prrof, you can take a look at the serie of papers by Lihe Wang where boundary regularity is also investigated.
http://onlinelibrary.wiley.com/doi/10.1002/cpa.3160450202/full (SECTION 4) http://onlinelibrary.wiley.com/doi/10.1002/cpa.3160450103/abstract