What determines which frequency range a radio telescope has?
A parabola does indeed focus over a broad frequency range. The lower limit is determined by the dish diameter, the upper by the construction (mesh size, parabolic accuracy etc.).

The collector placed at the focal point may be a simple dipole or other fixed frequency antenna, or more commonly, a waveguide that leads the collected signals to a low noise amplifier and receiver. The physical dimensions of the antenna or waveguide set the operating frequency range of the antenna system. More than one range may be tuned by means of multiple switched collectors.

The receiver is fundamentally no different to any other receiver; a tuning element selects the signals of interest which are then detected and amplified.