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