As a meaning of "sulk", 삐지다 is a dialect of 삐치다. past tense of 삐치다 is 삐쳤다. past tense of 삐지다 is 삐졌다. So 삐졌다 is a dialect of 삐쳤다. NB 삐지다 has an another meaning of "slice" 칼 따위로 물건을 얇고 비스듬하게 잘라 내다. which is a standard language.

So the correct form is 삐치다 삐졌다 or 삐지다 is a wrong way of using it. And 삐쳤다 is a short form of 삐치+었다(past form) To sum up, present from 삐치다 past form 삐치었다(삐쳤다) hope it would be helpful :)