It shouldn't be a factor in the Admissions process, unless the Applicant makes it one ...there is an opportunity to explain how you overcame adversity et. Once of greatest things about USNA is the diversity of the class...sure, there are kids who grew up with a silver spoon, best schools , etc., but there are plenty who just came from the farm in rural Timbuktu or grew up in single family homes. As someone pointed out, one of the goals of Admissions (and Navy Officer accessions programs in general)., is to make the officer corps reflect the sailors we lead. It's not perfect, largely because that sharp young kid from a troubled background doesn't think he/she can get admitted... As I have often said here, 100% of those who don't try won't get admitted.
JustDoIt did mention the impact on a security clearance., As a general rule, I would not expect that a young officer would be held back by the sins of their Parents, particularly if they are open and honest about it if asked. Background investigations for Security Clearances can be very probing (depending upon the level of clearance required for your position) The objective is not to make the officer uncomfortable, but rather to identify any issues that an adversary could exploit in order to cause a compromise of confidential information. For example, is there a deep dark family secret that would cause the officer to choose to compromise information in order to keep the matter secret. Like other aspects of life, honesty is the best policy, and if you are upfront with the investigator, a parent's background shouldn't impact security clearances.