June itinerary

R Steeves mentions Hydra as an island that can be visited from Athens for a day or two when you don't have time to go to another island. In your case it will be of no interest (and rather a waste of time) since you will spend a lot of time in the islands
So take Hydra out of your schedule.

Depending on the arrival time of your flight in Athens you can just have a stopover in Athens airport and find a flight for example to Santorini (with Aegean/Olympic or Sky Express).
This saves you having to pass through Athens twice and you will be in Santorini from the 1st day.

Note that if you are flying with a Skyteam Alliance airline (United, Air Canada, etc) you might find a codeshare flight with only 1 ticket "Home-Island via Athens" since Aegean/Olympic is part of this alliance

In case your flight arrives too late in Athens, stay overnight in the port of Piraeus and take a ferry the next morning.

The other good habit when traveling to the Greek islands is to always keep Athens for the end of the stay and not to return there on the day of your flight back home because the ferries or even the flights from the islands can be canceled due to breakdown, strike, and especially bad weather conditions (even in June).

Staying 2 nights in Santorini, maybe 3 nights is in my opinion a maximum if you do the same usual tourist circuit that the millions of other tourists do. (Fira, Oia and its so-called famous sunset photographed millions of times, Akrotiri, etc.)

Athens deserves at least 3 or 4 nights.

So even if you stay 3 nights in Santorini and 4 nights in Athens, you would have 9 nights left for Paros, which may be too much for you.
Knowing the other members of this forum, I'm sure they will advise you to add Naxos and to distribute these 9 nights between Paros and Naxos.
It's a possibility. There are many other options but if you want to do like everyone else, why not.

In any case, consider rather a schedule like this:

1/ Fly to Athens - Layover in Athens & fly to Santorini.

2-3-4 Santorini

4/ Ferry to island A

4-5-6-7-8 Island A (for example Naxos)

8/ ferry to Island B

8-9-10-11-12 Island B (for example Paros)

12/ Ferry to Athens

12-13-14-15-16 Athens

16/ Fly home.

In June you will have several ferries a day to travel between these islands and back to Athens.

Ferries timetables and prices will be published around mid-March and can be consulted for example here:
https://www.ferryhopper.com/en/

Thank you!

Instead of visiting 3 islands, a nice mix is to also visit 2 islands and the Peloponnese Peninsula. You can stay in Nafplio and visit some amazing ancient sites, such as Epidauros, Mycenae, Mystras, and others. And visit Athens at the end. This was our itinerary last May:

Santorini - 3 nights
Naxos - 4 nights
Nafplio on the Peloponnese Peninsula - 5 nights
Athens - 4 nights

This is a very common itinerary for good reason. So it is not off the beaten path. There are many other islands but I don't know enough about them. Paros would be a good choice, too. When I was planning our trip I initially included Paros in addition to Santorini and Naxos, but after doing more research, I decided to omit Paros and add Nafplio instead.

I don't know where you are flying from. We flew Boston-Zurich-Santorini on one ticket on Swissair. If we missed our flight to Santorini, Swissair would be responsible for putting us on the next flight. And we didn't have to worry about our luggage until we arrived in Santorini. Then we took the Blue Star Ferry from Santorini to Naxos, a 2-hour ride. Then we flew on Aegean Air to Athens airport to pick up our rental car and drive to Nafplio, about a 2-hour drive. And flew home from Athens after spending 4 nights there.

I do agree with Jolui about your itinerary from Day 11 through 16. Too much traveling around for very little gain. I visited Hydra many years ago. It is fine if that is the only island you are visiting. There are so many more interesting islands than Hydra. And it's a long taxi ride into Athens from the airport. So your days 11 & 12 are a total waste. Too much going back and forth between Athens.

Thanks for replying! We have a party in Paros so have to be there on those dates. We are flying from Israel so not so far

Since you have to be in Paros from day 5 to 7, see if you can take an Athens-Santorini flight on day 1, spend 3 nights in Santorini and on day 4 take a ferry to Paros.

After Paros you could also spend a few days in Sifnos which is a beautiful island, quieter than Naxos and much more authentic than this kind of theme park / Disneyland that Santorini has become.

https://sifnos.e-sifnos.com/

The Paros-Sifnos trip takes 1-3 hours depending on the type of ferry

Then return to Athens by ferry and do as kmkwoo suggests, instead of staying in the islands, go to the Peloponese (hire a car) and return to Athens for the last 3 days.

Thank yo7 so much for replying. We’ll check this out