Go with 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 it’s short and heavy enough (12 gauge or better), but you’re cutting it too close. Compressors have a startup surge — that inrush current can easily spike 2–3 times the running amps for a split second. The 15-amp cord might trip a breaker, or worse, the wire heats up and degrades over time.

The real issue is distance. A 15-amp cord rated for 14 gauge is only good up to maybe 25 feet at 13 amps. Go longer and voltage drop kicks in, making the motor work harder and possibly burn out. A 20-amp cord (12 gauge minimum) gives you a safe buffer for longer runs and startup spikes.

The price difference is usually a few bucks. Don’t save five dollars and risk ruining a compressor or starting a fire.

Compressors are hard starters — give them the headroom.