We've definitely got a lot of different genres at work here, and they all interact together in some interesting ways. First off, we have comedy. This book definitely portrays its characters humorously at times. The narrator often ridicules people and gives us witty observations. We also have lots of absurd and humorous situations that happen, many times resulting in embarrassment for someone. For instance, check out the scene in which Mary insults the clergy... just before learning that Edmund is about to become a clergyman (9.26).
But there are very strong elements of seriousness in this book too. It's not a laugh-a-minute comedy. Funny moments exist alongside very painful and sympathetic ones. In fact, the only character consistently ridiculed is Mrs. Norris, and even she's treated with some sympathy by the novel's end. All the other characters are involved in very dramatic moments, many of which revolve around family issues, and that get serious, sympathetic treatment from the narrator.
The family drama and character shenanigans are also offset by a contrast between the city and the country (check out the "Setting" section for more on this). The country is often romanticized and depicted as better than the town. This romantic depiction of the country and of country life means that Mansfield Park definitely has some elements of the pastoral. Though this isn't the dominant genre at work, it's still worth noting.
The last genre we have is romance, and this one is really the major umbrella genre of Mansfield Park. Romance is at the root of both the comedy and the drama in this book, and most of the major plot threads and themes are related to romance in some shape or form: marriage, falling in love, flirtation, unrequited love, etc.