Actually, the technology to do something similar to this exists today, although at a smaller scale (and though the technology exists, the political will does not). You don't need many black holes, necessarily, but they could help (I'll get to this in a bit).
Right now, we call this technology FOCAL (short for Fast Outgoing Cyclopean Astronomical Lens, which seems like a bit of a stretch acronym to me). The basic idea is just what you say, to use the gravitational lensing of a large mass (in this case the sun, in your case a black hole) to focus light. The lensing allows you to focus on whatever is exactly on the other side of the star from where your observer is. There are some limitations (e.g., the focal point is about 550 AU away from the star, the corona of the sun will mess up the image, and each FOCAL satellite will only let you view one spot in space), but for a civilization that can shuffle black holes around these shouldn't be a problem. Anyway, this tech is promising: even the (relatively) small gravitational field of the sun is theorized to be a good enough lens to allow us to view the surfaces of exoplanets in detail.
The Advantages of Black Holes
There are really two main advantages of using black holes over using stars. First, they're dark. If you get a black hole that's not ejecting lots of mass and doesn't have a big, bright accretion disk, you should theoretically end up with a much less distorted image than if you are using a bright start that's constantly shooting plumes and loops of plasma into your viewing field.
The second big advantage is that they bend light much more strongly, which (if I've understood the optics, although it's entirely possible that I have not) will reduce the distance from the black hole that you need to be to get proper focusing. As long as it doesn't reduce it too much (i.e. as long as the focal point is outside the event horizon) this makes the entire process easier.
Next Steps with Kardashev
There are two big improvements you could make if you have access to the type of power a KD 2.5 civilization would wield. First, for any Type 2 or 3 civilization, making many observers for each black hole would allow for much wider coverage of the sky. I don't know how you'd work out the mechanics, but if the focal orbit is close enough to the black hole you could probably cover the whole sky by having the observers orbit the black hole and collect data constantly.
The next advantage, probably only open to a type 3 (or high type-2) civilization would be to arrange multiple black holes - as you posit in your question - to exploit the parallax to generate pseudo-3D renderings of observed objects. If you observe from multiple, close together spots, you get a depth perception effect like what humans get by having two eyes. If you observe from multiple, far apart spots you could theoretically get true-3D information about an object, although this would only work for objects inside the galaxy as you need to have black holes arranged on multiple sides of an object.
Additionally - and it would take someone very skilled in both optics and general relativity to know for sure - it may be possible to arrange the black holes in such a way as to achieve even greater magnification. My naive thought is that one black hole placed on the intersection of the focal lines of several other black holes might achieve this, but really that's just wild speculation.
Obviously this is a huge undertaking, and without the benefits of FTL communication it may be impossible to properly coordinate. The other aspects of your built world are going to play a large role in how or if this is achievable.
Also, keep in mind that focusing on and looking at something really far away is equivalent to looking at that thing far back in time. If you really could achieve telescopes like this, you would be seeing events anywhere from thousands to billions of years ago, potentially (if the math works out) observing the details of the very early universe, or (with less powerful telescopes) the births of the first stars/planets/civilizations. Worth keeping in mind for plot purposes.