I was able to access the link at 10:00 PM Pacifuc time - so as I suspected, it's similar to the websites for other tourist sites in Italy, which are only available during Italian business hours. So depending on you time zone, try accessing the link early in the morning if on Eastern time, or late in the evening on Pacific time.
Here is the link to the ticketing site, which is a third party - VivaTicket - there's a chance it may be accessible at all hours:
https://www.vivaticket.com/it/ticket/parco-archeologico-di-paestum-velia-durata-3-giorni/171405?culture=en-gb
Here are the instructions for buying tickets from the Park's website:
ONLINE SHOPPING GUIDE
To buy online the entrance ticket to the Archaeological Park of Paestum and Velia:
GO ON VIVATICKET
Before starting the purchase procedure, log in or register on the Vivaticket website.
Choose “Subscription” by clicking “Buy” and follow the payment procedure.
At the top right there is the search bar: in this space type “Paestum”, here a dropdown opens which allows you to choose between an annual ticket and the daily ticket (choose “Archaeological Park of Paestum and Velia”) valid for 3 days that allows you to visit the Park of Paestum and Velia;
By selecting the option “Archaeological Park of Paestum and Velia” there is the possibility to choose between the full-reduced ticket or family ticket and also the time slot for booking the visit in the two archaeological sites.
With the purchase receipt you can enter into the Park without passing again from the ticket office.