In Paros with kids aged 8 and 12, check out Aliki. It's in the south of Paros. There are two beaches sheltered from the wind. It's a village with restaurants, a mini market, a few shops, and a pharmacy. It has a rather family-friendly vibe, and there are quite a few accommodations for families (small apartments or houses).
https://www.youtube.com/watch?v=5tAh5smWQow&ab
With slightly fewer facilities, check out Piso Livadi, to the east. It's in an area with large beaches, including Golden Beach, the largest in Paros. It's only sand until the small village of Drios. You're facing Naxos, which you can see on the horizon.
https://www.youtube.com/watch?v=EbGIXGx-8O0&
It's less sheltered from the wind, but it allows for a lot of watersports activities such as windsurfing, kitesurfing, etc. There are some nice clubs, even for beginners, for example:
https://force7paros.gr/
Throughout this area, there are beachfront accommodations, from studios to villas with pools. There are some good restaurants and bars on Piso Livadi and Drios. Anna's restaurant in Drios is one of the best on the island; it serves only home-cooked food, and I'm not just saying that because Anna is a friend :).
https://www.restaurantannaparos.gr/
Otherwise, if you're looking for a more lively atmosphere, even very lively at times, with all the amenities of a small town, you can always check out Parikia, the capital.
These are just suggestions, not recommendations, since I don't know how you spend your vacation.
But still, Naoussa doesn't seem at all suitable for a family with two young children.
I've gathered some information about Paros on this map:
https://www.google.com/maps/d/edit?mid=1z9zcPtVkkjzEwrXDj6XUG_5n_jcNaYZb&usp=sharing
I don't know your budget or accommodation preferences, but if you're a beach lover, here's an example of a place to stay in Paros.
Near Piso Livadi, the house is the one in this Street View image. https://maps.app.goo.gl/zS1SvN4wpKCc22hS6
It's the same one with the blue shutters that we see at 0:20 of this video: https://youtu.be/8oDrypOkxcw
It has two bedrooms and costs around €160/day in June.
https://www.airbnb.com/rooms/10678113