Parents can influence the decision by how independent they will force their child to be. If a parent is unhappy that an 18-year old who has unilaterally decided to not attend college but work at the local grocery store and buy a new car, that parent might decide it’s time for that child to live on his own and enjoy all the benefits of being an adult. A parent may decide it’s a learning experience to have to figure out how to pay rent, utilities, feed oneself, cell phone, car note, etc, out of $10/hr.
If a child is faced with living completely on his/her own with absolutely no parental financial assistance, or go to college as the parent desires, the child may feel the decision has been taken out of his/her hands.
IMO, while I would not say the parent has 100% control over the decision, cutting off all financial assistance moves the needle away from the child having 100% control over the decision. Again, in my opinion, very few middle-class children are going to decide to live in poverty in order to avoid attending college.