Consider the line tangent to y^2=4x. It has m = x^(-1/2). At 1,2, that's m=1, so y=x+1. The circles are tangent to this line, so a perpendicular through (1,2) joins their centres. That'll be y=-x+3.
So to get our centres we're looking for two points with a distance of 4 from (1,2) along y=-x+3.
Since the slope is -1, the x-distance must be the same as the y-distance, so the distance in both directions must be root(2).
Given that, you have your centres, and plugging that and the radius into the standard equation of a circle finishes the job.