I’ve been reading in the TripAdvisor forum that the trails are still
closed. So I went on the park site and they are listed as closed
Only SOME of the trails are closed. If you look at a park map, you'll see that there's a pretty extensive trail system within the park and the majority of the trails are open.
http://www.parconazionale5terre.it/Esentieri-outdoor.php
The route that is (or has been) closed is also the most popular: the 4 legs of the SVA2 - or Sentiero Azzurro - between Monterosso and Riomaggiore. Two of those legs, between Corniglia>Manarola and Manarola>Riomaggiore, have been closed years and will likely remain closed this season; Manarola>Riomaggiore for sure.
I see that the Monterosso>Vernazza section of the SVA2 is now open (and rated as "skilled"), and Vernazza>Corniglia will likely be open sometime this spring as well unless it falls, or has fallen, victim to some sort of damage which would take considerable time to repair.