Does Islam require that I get married because my parents say so and because it would make them happy even if I am not happy or ready?
No, Islam doesn't require you to get married because your parents say so. Moreover, arranged marriage is just a popular tradition, and has nothing to do with Islam. This tradition is widespread within non-islamic countries too. It is up to you to chose if you want to follow your country's traditions or not.
If you chose to break this tradition, try hard to be polite and nice to your parents. Be respectful and extremely careful.
According to Islam, there is nothing wrong in getting married at 32 or 42, it doesn't make you a bad or non-religious person, it has nothing to do with your age.
In Islam, you have to be very respectful to your parents and your family. That doesn't mean you have to follow everything they say. Don't ruin your family relationship while following your heart.