Inspired by a Snowman.

Let the radii of the two circles of set $A$ be $r_{A1}$ and $r_{A2}$ and similarly for the remaining two sets. Then we note that $2\left(r_{A1}+ r_{A2}\right)$ $= 2\left(r_{B1}+ r_{B2}\right)$ $= 2\left(r_{C1}+ r_{C2}\right) = h$ , where $h$ is the vertical distance between the top and bottom parallel lines.

Since the perimeters of the three sets of two circles are: $p+A = 2\pi \left(r_{A1}+ r_{A2}\right) = \pi h$ , $p_B = 2\pi \left(r_{B1}+ r_{B2}\right) =\pi h$ , and $p_C= 2\pi \left(r_{C1}+ r_{C2}\right)=\pi h$ . $\implies$ They all have the same perimeter.