"Mulligans" is a term in golf, which means "have your shot again" (after making a mistake). The author is using it as an analogy to mean "I allow my kids second chances when they do something wrong."
"Text messaging" refers to using a mobile phone to send SMS messages to friends ("texts").
"Swimsuit shoots" are modelling events involving women or girls wearing swimming costumes. I presume he means that he takes his (teenage?) daughters to events where they model in these costumes.
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