How do I remove a stubborn exterior threshold?
That threshold is part of the entire door system. It wasn't added after the door was installed. It's screwed upward into the jamb sides and may have extrusion fins that engage slots in the jamb. Ordinarily a threshold doesn't wear out before the door system does, so it's rare that a person would try and remove it without replacing the door. 

You'll need to figure out how to cut those screws so you don't tear out chunks of jamb. It's made from soft pine and would splinter fairly easily.