Shouldn't it be Παρακαλούφη instead ?
Well the term is infact correct but in modern Greek you wont hear anybody say παρακολουθώ τηλεόραση
They will say βλέπω τηλεόραση i see tv instead of watch
is 'κοιτάζει' wrong?
Yes, because κοιτάζω is the equivalent of to look at. Unless you're looking at the TV (κοιτάζω την τηλεόραση), the actual device that is, you'll never use κοιτάζω for that activity.