I am planning to install 3/4" hardwood planks. I would like to run them parallel to the floor joists. The existing subfloor consists of wooden planks laid diagonally to the joists on 16" centers. Since the subfloor is diagonal to the joists can the new floor be laid parallel to the joists without any additional reinforcement? Also, there is a fireplace in the room- should that be framed?

It's o.k to run the floor parallel to the joists , depending on the situation. Sometimes it's a flatter installation running opposite the joists, ( depends how flat the subfloor is between the joists). Also a tough call without knowing how firm the subfloor is between the joists.

I usually frame out fireplaces, basketweaving the corners.

If the hearth is going to end up flush to or nearly flush to the hardwood, a frame finishes it off very nicely. If the hearth is elevated, then I generally don't frame it. It's your call since it is a matter of aesthetics, not structure.