First link does not work.

Sorry about that, here's another link http://bonavista-apartments.com/Establiment.aspx?codi=1005&b=n

The bona vista is in the most central location
The booking.com is the worst location

1) The Bonavista apartment is in a very fine location. You can literally trust the wording on its website when you click on "The Area".
The main appeal would be its proximity to upscale upper Passeig de Gràcia, Rambles de Catalunya and neighboring streets of Eixample, the famous Gaudí and other Modernisme mansions in that area, and to quirky Gràcia. All in walking distance. Plus metros 3 and 5 on your doorstep at Diagonal for fast connection to Pl. Catalunya, the Old Town, and Sagrada Familia.

2) Hotel BCN Parc Güell would be too far away from anything. Not a bad neighborhood at all. Just somewhat boring.

3) Airbnb's Gràcia apartment is a bit special-interest. Gràcia is a very nice, small town like neighborhood with many restaurants, shops, etc. But you will need to take the metro at least two stops to get to the big sights of Barcelona.
You can walk to "quaint" Gràcia, though, from the Bonavista apartment.

So I would rank them
#1 Bonavista, hands-down winner
#2 Gràcia, an interesting alternative with a different flair
#3 Parc Güell, well, not even #3 but not really in the same league as the other two.

The Bonavista one is the best, the others suck.

Parc Guell is near NOTHING of tourist interest. It is far from the center of town and not convenient to the Metro. It's great to visit, but you set aside time to visit it from your temporary residence that is more centrally located, not the other way around.

Don't rent from airbnb.com. There are horror stories on the 'net and you have ZERO recourse if the lessor is a scam artist. You want to rent from someplace reputable, check out the listings from www.apartur.com, which aggregates Barcelona apartment rental companies.

Thanks for the advice. We were torn about staying in the Barri Gotic or elsewhere, but does this seem like a good central location (Bonavista I mean) for visiting the various areas of Barcelona?

Yes.