Actually it's the other way round. When conjugating du-form you usually add ending -st. When the root of a verb ends with s, ß, z (lesen, gießen, sitzen) you leave out the s from -st because you could not pronounce the (second) s in -sst, -ßst or -zst. You can only pronounce -st, -ßt or -zt. So we leave that out in writing, too. We have a similar problem when the root ends with t or d. Try to properly pronounce -tst and -dst. Real tongue twisters! Put an e in between and you don’t have any problems anymore!
In principle the same applies to Präteritum. Usually the ending is -st, if it’s more comfortable to pronounce you may shorten it to ending -t or put in an e and get ending -est. The good thing about Präteritum, except for modal verbs (which you can learn by heart) we rarely use it in spoken language and in writing you only need it when telling something in narrative style.