Can I get parasites from uncooked vegetables?

Yes. It is generally advised that you rinse all fresh produce before eating it. If you are still concerned that it may be contaminated, cooking the food item can kill possible parasites and bacteria. Do not contaminate foods by using knives, counters or cutting boards that may have come in contact with other infectious sources such as raw meat or poultry.
Yes, it's possible. Raw fruits and raw vegetables have many benefits, but there are some drawbacks, which is why all human cultures eat cooked food. Raw plants have tough Cellulose fiber, so require more chewing to break into the cells so we can get the nutrients. Undigested fiber is fermentation food for our gut bacteria, leading to gas. Without cooking, one eats live bacteria, fungi, and parasites that are on food.