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

The result holds true in the parabolic case for F uniformly elliptic.

For a proof, you can take a look at the series 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

See also https://projecteuclid.org/euclid.bams/1183555461 and the very interesting book of Krylov http://www.springer.com/us/book/9781402003349.