How to construct a door header without enough height?
I see several options, though it's difficult to be sure from one photo.

Set the header into the roof framing cavity. You mention that the rafters are parallel. If they're not directly over the header you may have more room than you realize.
Since the wall is parallel to the rafters you may not actually need a load-bearing header. Doubled 2x6 may be adequate. You can possibly fit that just by removing a wall plate.
Have a steel beam sized by an engineer. Your local lumber yard can probably help with that, and they may not charge extra.