Cars are allowed up to a certain point which is the parking lot of the Park where free parking is available.

The Monastiri Beach is a public beach open to all. A segment of the beach does have sunbeds and umbrellas for customers who wish to rent them for the day. But this is optional. There is plenty of free space for visitors who do not want to hire them.

Yes, there is for July and August.

Tickets for the “Festival at the Park” events can be found On-line and at the Café Parko, where credit cards are accepted: open daily from 10:00 to 19:00.
In Naousa during shop hours: photo-store Wide-Angle (main square), tel: +30 2284 051933.
In Parikia during shop hours: book-store Anagenisis (ekatontapyliani str.), tel: +30 22840 2284022247 and book-store Polychromo (Ring Road), (tel: +30 22840 22906).
Phone ticket reservations are also available at the Cafe Parko (tel: +30 22840 5356). Tickets can be payed by credit card and collected on the day of the event.
Tickets are also available at the entrance to the Park on the day of the event, usually at a slightly higher price.

Paros Park offers free WI-FI to guests of the Cafe Parko.

Absolutely!

The Park is open 12 months a year. There is free pedestrian access to the walking paths, the Archilochos Amphitheatre and the beaches of the Peninsula. The Cafe Parko however is open from June till September.