Medial elbow pain (pain on the side of your elbow) is common with athletes performing high volumes of muscle-ups, rope climbs etc.
Just like last week with pressing movements, 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 do more pulling with a bia for producing external rotation (face pulls) or pulling movements that subtlety resist internal rotation (X-pulls).
This means you can keep your training volume relatively high while letting the injury heal.
Again, just like last week (Reverse Grip Pressing #151) when we talked about medial elbow pain and pressing movements, 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.