Following directions on pairs in exploding kittens?
As per the VERY first paragraph in the "Special combos (for advanced play)" section on the rules:


  When you play combos, ignore the instructions on the cards and do the following instead:


Therefore, in the situation you presented, you would still be required to draw a card to end your turn as you played a Three of a Kind, not a skip card. If you had the remaining skip card, you could play it separately to end your turn without drawing a card.