The main difference between the two is the process that they do to get their food. A mushroom is considered to be a saprophyte, and this is in charge of getting food from other plants so that it will thrive and survive. A parasite can get food from both plants and animals.
For example, a tick may live on a dog for the longest time because this means that the tick will have access to the dog’s blood, which it needs to survive. These two need to depend on other organisms to survive. If they would not find anything to latch on, their end is inevitable.