참다 is enduring or holding something back — tears, words, laughter, an urge. Repeating it across -고, as 참고 참다, is the standard way to say someone held out a long time before finally breaking, which is exactly how three of these caption lines use it (참고 참던 막내). 이 악 물고 참다 is the fixed phrase for gritting your teeth and bearing it.
Register: casual; often a caption over someone about to crack