There are lots of options out there. Many of which are available as downloadable software as well as public websites. I do not think many of them expect to be used as API's unless they explicitly state that.
The one that I found effective was Enju which did not have the character limit that the Marc's Carnagie Mellon link had. Marc also mentioned a VISL scanner in comments, but that requires java in the browser, which is a non-starter for me.
Note that recently, Google has offered a new NLP Machine Learning API that providers amoung other features, a automatic sentence parser. I will likely not update this answer again, especially since the question is closed, but I suspect that the other big ML cloud stacks will soon support the same.