What was Bradley's observation concerning abevation of light? As far as I understand, the effect of parallax exhibits itself in the shift of a star relative to the background stars on photographs. 

As for the aberration of light, how is it reflected? It would not cause the relative shift of a star in a background, right? Because all the stars (the foreground one and the background ones) will be rotated as a whole, right? 

So, what was Bradley's observation?
You are right that what you must use as a referent is different-- for parallax, your referent will be a much more distant object, because it shows little parallax.  In aberration, it happens to the whole visual field.  But what is shifted is the angle relative to the ground at which the light arrives, so if you set your telescope to see a given star, in principle you would need to correct for both parallax and aberration as the Earth orbits. But aberration is usually much larger than parallax because it scales with v/c rather than D/d (where D is 1 AU and d is the distance to the star).  For the Earth's orbit, v/c is about $10^{-4}$, but stars are much farther than $10^4$ AU.  As for Bradley's observation, a good version is given at http://cseligman.com/text/history/bradley.htm.  He was looking for the parallax effect, and stumbled into the much larger aberration effect.  He could tell it was something different because the sign was opposite and it did not depend on distance to the source.  This was 50 years after Romer, so the speed of light was already known.