Some people are allergic to them, so obviously that is a problem for them.
Using products with parabens consistently also increases the risk of certain types of cancer and can decrease sperm production in men. They can mimic hormones that are in the body (particularly estrogen) so that can throw things off over time.
Companies use them as preservatives in their products to give them a longer shelf life and also because it is less expensive than using other ingredients. It is one of those things where your dollar is your vote, and as long as people keep buying products with parabens, they will keep making them.