Today, I will need to wash the car, clean the air conditioner, and resealing the driveway.

Correct sentence: Today, I will need to wash the car, clean the air conditioner, and reseal the driveway.

One of the rules for writing with parallel construction is to ensure that verb tenses agree within the sentence. In this sentence, the verbs "wash" and "clean" are written in the future simple tense; so too should the verb "reseal" be in future simple tense.

At the doctor’s office, she said that he had a liver disease that, while not life-threatening, should be cured as soon as possible with supplements.

Correct sentence: At the doctor’s office, the doctor said that he had a liver disease that, while not life-threatening, should be cured as soon as possible with supplements.

It is unclear, as originally written, who "she" and "he" refer to. "the doctor said that he" lets us know that the doctor told him that the latter had a liver disease.

We decided that our contribution to the game would be attacking and that they’re contribution would be defending.

Correct sentence: We decided that our contribution to the game would be attacking and that their contribution would be defending.

The possessive determiner "their" lets the reader know whose contribution would be defending. In this case, it is some group of people. "They’re" is a contraction of "they are" and is inappropriate and illogical to use.

Rachael was able to prove to her sister, Amanda, that her’s was the right way to cook the soup.

Correct sentence: Rachael was able to prove to her sister, Amanda, that hers was the right way to cook the soup.

The possessive pronoun "hers" is used to refer to Rachael’s way of cooking the soup. There is no apostrophe needed to indicate possession.