As far as honorific speech is concerned, your own parents are not any higher than yourself in status. You treat them as your equals when speaking to a third party. If you have been taught otherwise somewhere, it is indeed unfortunate. That is why you must say 「[親]{おや}を～～に[連]{つ}れていく」, instead of saying it using a "better" verb that will be introduced below.
If, however, you are taking another person's parents (outside of your family), your teacher, your customer, etc. to a place, then you will use 「～～を～～にお連れする」. Using 「連れていく」 in these cases will make one a very poor keigo-user (and in the business world, you will be called out).
Point is it does not matter how much you personally respect and admire your parents. That is your business and it is cool. You just do not elevate them as objects of respect when speaking to a third party about them in keigo. This is extremely important and is surely a weak point for many Japanese-learners.
「[両親]{りょうしん}を[旅行]{りょこう}に連れていく」
「親を[病院]{びょういん}に連れていく」
Both phrases above are perfect (because the speakers are not elevating their parents in status).
Again, you should never, ever replace 「連れていく」 by 「お連れする」 in the phrases above. That would be "trying to" speak politely and failing miserably. "Comical" is how that would sound to native speakers.
「連れていく」 is for people lower or equal in status, not just lower as you stated. Your family members are your equals in the keigo world.
「お連れする」 is for people higher in status. In reality, however, the politer speakers use it even when talking about taking a stranger somewhere if the stranger is not way younger than themselves. But they sure will not use it for their parents or grand-parents because they know they should not.
