Yes, eventually all parking will be in structures and the hospital will expand across Fourth St. The plans are for a building to bridge the Fourth Street right-of-way on the second and third levels, while ground level remains open to either just peds/bikes or everything depending on how that discussion turns out.
The fact is, parking structures aren't cheap to build, and they're not going to build them until they run out of surface room. The lone "future parking garage" is completely nebulous at this point...no design and no idea when it would be built. It's positioned there to be consistent with the overall vision for the site, allowing it to remain there and be expanded to fill most of the future Owens Street frontage when the hospital expands. Build it closer to the first-phase hospital and they'll have to tear it down when they want to build the second phase, which may come sooner than many think.
There is a park in the SW corner of the parcel that will be developed along with the hospital. It is currently scheduled to be a soccer field, but some neighbors are questioning whether that's what they really want to see there.
For reference, this is a concept for full build-out...plans have evolved since this was put together, but the basic idea remains the same.