Hi! We arrive in Athens (from Boston) on August 25 at 12:45 PM(unless Swiss Air changes the timings). We plan to stay in Antiparos on night one. Looking at various flights - it seems like we can book flights to Paros from 3 PM onward. I am calling trip insurance in a few hours to find out what I can do if our flight times get changed (this has happened before!) Or is it best to play it safe and overnight in Athens? I'd rather not do that...

I am guessing I could just book the flights and rely on trip insurance if things get weird. If Swiss alters its flight schedule, perhaps I could negotiate with them to fly one day earlier?

I don't think that Swiss will change their timetable.
The problem is rather that the airspace over both, Switzerland and Greece if often overcharged in the summer holiday season and that the local flight from ZRH to ATH can get late.
May be you book a 4pm flight from ATH to PAS.

There are flights every hour it seems after 4. Travel insurance won't cover unless there is a 6 hour flight delay.

I wouldn’t worry too much about the Athens - Antiparos end of your connection. Book Flexible tickets from Athens to Paros, and if you can’t make the 3 pm flight and later flights are full you can still catch the 5:30 pm Blue Star from Piraeus. It will arrive on Paros around 9:30 pm, and ferries from Pounta to Antiparos run until 02:00 in the morning in August. Antiparos hotels are used to late arrivals, as many people use the 5:30 pm Blue Star to get to the island.

It may be exhausting but maybe we should just take the 5:30 PM ferry from Piraeus.

Overlt, don't give up on the flight just yet, although I would be inclined to book the 4 pm rather than the 3 pm. With a Flex ticket on Olympic Air (part of Aegean Airlines) you can change your ticket without a rebooking fee:
Olympic Air Fare Rules

It's actually easier to reach Antiparos from the airport than from the ferry port, so very convenient to fly. I mentioned the 5:30 pm Blue Star as a last resort if you have to rebook and the later flights are full. If you do take the 5:30 pm Blue Star, recommend buying business class tickets or even a cabin if one is available. Cabins will probably bet booked up quickly, though, because the ferry goes on to Amorgos after stopping at Paros, arriving there quite late.