Should I use a 20-amp extension cord for a 13-amp compressor?

Yes, get the 20-amp cord. A 15-amp cord can handle 13 amps continuously if short and heavy enough, but compressors have a startup surge that can spike 2–3 times the running amps, risking a tripped breaker or wire degradation. A 20-amp cord (12 gauge minimum) gives a safe buffer for longer runs and startup spikes, and the price difference is minimal.