Medial elbow pain (pain on the side of your elbow) is common with athletes performing high volumes of muscle-ups, rope climbs etc.
Once the injury sets in and starts hurting, it can continue to be aggravated when doing almost any standard upper body pressing and pulling movements (pushups, rows etc).
One way to work around this (mild) pain is to switch to a supinated grip so less tension/stress is taken on by the medial side of the elbow and therefore does not exacerbate the issue (as much).
This means you can keep your training volume relatively high while letting the injury heal.
Granted, if the pain is extreme then these movement variations may not be enough to remove the pain and seeing a physical therapist may be best.