Many UK operatories are pretty small too although not all. Space is just an excuse used in USA. A relative can stand in the corner or sit on a chair in the corner. In UK for medical not dental, I was with my child until he was asleep from GA and then I left him, the actual OR is the only place you cannot go, although if delivering a baby under local that is also allowed for the father to be present in the OR. They have to put the gear on like the staff.
If you are still under 18 in USA, there is no law to prevent your parent 'going back' with you but the dodgy 'medicaid corporate chains' are well-known for citing the HIPAA laws as a reason. This is not a valid reason at all.
Since you are deaf I think you should pitch it that your Mother's presence is a benefit to the dentist as she can facilitate communication and that it also helps keep you calm. Have your mother make the appointments, come along and simply behave as if she is going nowhere but with you...in other words don't succumb to their stock phrases too easily...'say bye bye to Mom now....'
In fact any dentist who won't let you have your emotional support along is not really the dentist for you. Just because they prefer you to be alone doesn't mean you have to be alone.
If you are going to a medicaid corporate dental chain, then please reconsider as the chances of them giving you a good experience and appropriate treatment are far less than if you go somewhere reputable. At least do a google of reviews before you go there.
Good luck - there are plenty of dentists out there, no need to stick to the one you have always used especially if they have left you feeling scared.