Ακολουθώ means "Follow". e.g. Θα σε ακολουθώ, καθώς περπατάς, για να μην χαθώ = I will follow you, while you are walking, so that I won't get lost.
Παρακολουθώ on the other hand has those meanings: "Watch (carefully)". E.g. Παρακολουθώ τηλεόραση = Ι am watching TV.
"Attend": e.g. Παρακολουθώ το μάθημα = I attend the class.
Another meaning is "Have somebody under surveillance": E.g. Η αστυνομία παρακολουθεί τον ύποπτο = The police has the suspect under surveillance.
"Stalk" eg, Ο Γιάννης είχε τέτοια εμμονή με τη Μαρία, που την παρακολουθούσε όταν αυτή έβγαινε έξω. = Giannis was so much obsessed with Maria, he was stalking her whenever she went out.
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Vasiliki Baskou, Instructor/Director, https://learn-greek-online.com.