If your Dad's trainer cannot do it, at his age he needs to be medically cleared anyway, so I would have him go to his doctor and see about getting (along with his clearance to exercise and any other notices to his trainer, which the trainer should have sent him for already) a referal to a Physical Therapist who may be able to do a more in depth posture and gait analysis. It may be that, if the trainer recommended it, he saw something off balance that he thinks your father needs to work on. Without seeing your father I cannot recommend any particular fitness test, nor would I do one without his doctors consent.
Good luck