We cannot see the supernova remnant of the star that triggered the formation of the Sun and the Solar System, as the remnant dispersed and became mixed into the interstellar medium. So I am wondering, what is the maximum size/age of a supernova remnant/planetary nebula can reach before it can no longer be distinguished?
Of the order 10000–100000 years. As an example of an old Supernova remnant, you can see the Vela SNR, estimated at 11,000–12,300 years old.
Nasa describes the evolution of a remnant
Initially, the shock wave expands freely, As it grows it sweeps up material from the interstellar medium which slows its expansion. Expansion causes adiabatic cooling (or the Sedov-Taylor phase) until the point that electrons start to recombine with nuclei. This then allows for more effective thermal emission of UV radiation, which cools the remnant further, and it "disperses into the surrounding medium over the course of the next 10000 years."
A paper (mentioned by Rob in comments) describes this in greater detail, and offers this graph of the radio luminosity of a supernova remnant.
Illustration of a model radio light curve for Type Ia and CC SNR, evolving through the ejecta-dominated and Sedov-Taylor phases. The dashed line marks the transition to the radiative phase, where we assume the synchrotron emission becomes inefficient,