Your best chance to find something for €50/night or less will be by using the filter system at www.booking.com
I just made a trial booking inquiry for Paros for September 21-28 and focused on the main village of Parikia. There were several that popped up, including one that I stayed in a few years ago, Alexandra's Rooms. The beach is right across the street. The room was bright, clean and had fantastic water pressure and LOTS of hot water. There was a fridge, hot plate and coffee maker. I doubt you'll find a stand-alone little house with a full kitchen for €50, if that's what you're looking for. I also suggest you look at Aliki on Paros, and perhaps the island of Antiparos as an alternative.

I"ll bet that offer you to work on a fishing boat was not in Greece ... those fishermen go out long before dawn, and it would not be a treat, their boats are small and they work hard, not a tourist thing.

One nice thing about Naxos is that there is a company based Naxos/Paros that offers quite a variety of day excursions ... https://visit-cyclades.com/day-cruises-naxos-cruises/ browse their website, they have schooners that cruise to the small islands south of Naxos, or to Antiparos .... if you want a solo charter boat with skipper, you can e-mail this company and ask about availability.

do not forget to visit antiparos

have a wonderfull vacation

Yes, you can still find rooms on arrival in Paros town/port, Naoussa and Antiparos.
There is no problem with that. Depends what you want what it will cost. Your requirements are quite fussy and specific, rather hard to meet those.

As for islands being "party" islands that title is really for US must see shouty hives on maybe 3 of the central Cyclades.

I would have to go thru Paros town, Naoussa, Antiparos street by street to tell you where nightclubs and late music bars maybe located, this can, and has changed over the years.

I have been going to Paros/Antiparos occasionally for 30 years so an inaccurate guide book means nothing to me.
I have made mistakes booking remotely even though I know most islands, Rooms for Rent that used to be quiet years ago suddenly have two music bars next door.
If I had looked on arrival it would have been blatantly obvious.

I'll just do Antiparos as an example:
The main axis of music bars and some small night clubs is basically at the top end of main street where it jinks around to the main square and the main street to the left. The main square is noisy as you would expect, but in recent years is actually less noisy nothing like as packed as in many years ago. This probably reflects on the age of tourists which has gotten much older.
There is a "sometime" outside music bar a block or so left of main street, "sometime" as in some years it seems to be closed or seasonal.
There are a few Rooms for Rent/Apartments close to these areas but not many.

It's possible to find a room that way on Paros, but as far as I know the locals on Antiparos do not meet the small, local shuttle ferries arriving from Paros.

For a good room on Paros for the price you want to pay check Alexandra's Rooms. They're an easy walk from the Paros ferry port in Parikia. Turn left off the wharf and walk 750 meters.

http://www.alexandrasrooms.gr/pricelist

Santorini is NOT about beaches (they are black volcanic grit) ... you will have to go to other nearby islands for that. In high season it's impossible to avoid "busy tourist area" unless you go to far-flung islands that are hard to reach. You cannot have what you want at the time ou are going. However, there are an abundance of good sandy beaches on the nearby islands of Naxos, Paros, Antiparos. Sifnos has a lot of good beaches and is not as jammed since it does not have an airstrip --also, it has limited acccommodations, and when these fill up, that's it -- so the beaches never get super-crowded.

I would suggest starting with Santorini for TWO days only (because it will be hot & you'll want a swim, but you won't like the beaches, and hotels with pools will be €€€)... and then move on. Check out Antiparos ... is very walkable, to good beaches, and only a 15 minute ride on small ferry to the port town of Paros, and lots of nightlife. Then after a few days (depending on when you can book a ferry -- they don't go every day) go to either
(1) SIFNOS -. Beaches there you may like are Vathy and PLaty Gialos. Sifnos has SUPERB hiking trails

(2) NAXOS - Beach like Ag. Prokopios is large, long, deep water, good busses to town nightlife.

If going to Santorini, forget about Kef-Zak ... getting there involves too much travel time.

Another vote for Parikia and Antiparos. You can also get to Antiparos from Parikia, just not quite as frequently or as late into the night.

Sounds good, don't change your plans. 5 nights sounds adequate for Paros. The two main tourist villages on Paros island are Parikia (the port and capital; also called Paros) and Naousa in the north. While Naousa might be slightly more picturesque, I would suggest to stay in Parikia: close to the port, airport and the ferry to Antiparos. Frequent buses from Parikia to Naousa. If you feel comfortable driving, a 1-day car rental will allow you to see most of the rest of the island; highly recommended.

I also had 3 days in a wonderful Folegandros hotel the end of may, 5 years ago or so... but it wasn't a costly splurge: Hotel Anemousa - https://www.anemousahotel.gr/en/article.php?cat=2 has an "infinity pool" i think they cll it, right o the rim of a cliff ... I kept thinking I was in that perfume commercial ... at the time a single was €35 in may, now it's 45, but wowee! Only caveat ... you walk down a Steep ramp to reception & rooms/pool, with no handrail .. not fun at night (I hope they've changed that). I found that 3 days in FOlegandros was just right ... 3 days also marvelous in Amorgos (Aegali beach) and Serifos (Livadakia beach) ; Milos can keep u busy for 4 (car is needed here) , also Sifnos for 4-5 if you're a hiker ... Antiparos is a terrific relaxing stop for even 2 days.