Even if the public outdoor pools are open only in July and August, there is another option for water play during the hot days in May, June or September. Pack the towels, suncream and the change for your kids and head to one of the big parks. Most of the parks have a small lake or pond or jumping fountain officially used for water plays during the summer months.
Personally, we tried with our kids two places – the giant jumping fountain in the Seoul forest and the pond and jumping fountain in the Dream forest. Both places are amazing and the kids had lot of fun splashing in the water and trying to catch the jumping water jets.
Just keep in mind that the fountains are operating only every half an hour in the afternoons which allows your kids to have some break. Have a lot of fun and great splashing time!