In Paros to rent a car I usually go to either Inparos Rent a Car (inparos.gr) or Acropolis (acropolisparos.com)
They don't have brand new shiny cars but both are well priced and will be fine with your US license.
In case of unavailability there is also Cyclades Rent A Car (cycladesrentacar.gr)
The three are in Parikia the capital near the ferry port.
Anyway there is a very good bus service in Paros.
Most of the interesting beaches in Paros are on the east coast
The biggest and best known is Golden Beach but all the other beaches north or south of this one are fine (and you'll get a great view of Naxos while swimming)
If it's too windy to stay on the beach and you don't practice Kitesurfing, go for a walk in traditional villages like Lefkes, and from Lekes go to Prodomos and Marmara. There you will be right in the authentic Cycladic Greece.
In Prodomos there is an excellent little tavern in the square at the entrance to the village: O Tsitanis
https://www.facebook.com/tavernatsitsanis
In Marmara it is the Family Tavern Ververas where you have to go, even if the exterior is not very attractive. It is near the pharmacy.
And obviously, the only place not to be missed is in Drios, at Anna's, known to all the inhabitants of Paros and lovers of authentic meals.
Fans of fancy shops and expensive restaurants go to the village of Naoussa, which tries to emulate the vibe of Mykonos. It's pretty but has become completely artificial in recent years.
You might as well stay in Parikia, which has remained more traditional despite the floods of tourists in summer.
About hikes or walks you can go to the Paros lighthouse which is located in the extreme north west of the island. There are several circuits and possibilities to stop and take a break in very small coves not indicated on paper or Google maps.
Other hikes are possible in the center of the island by following the former Byzantine road (the vegetation has sometimes disappeared due to fires in recent years) or by going to the old marble quarries (the marble of Paros is one of the most beautiful in the world and known for its qualities of transparency)
Next to these old quarries there is a tavern in the tiny village of Marathi. The tavern name is also Marathi, it is frequented mainly by locals who come with their families to eat grilled meats, fries with bread coated in olive oil, Greek atmosphere guaranteed., English spoken not guaranteed :))
I stop because anyway in 2 or 3 days you will not have time to do everything and see everything