Newbie questions - Seeking a streamer with the right features - I'm confused!

You can stream to remote speakers via Bluetooth or AirPlay.

I am unclear if the "remote" speakers are going to be in a different room

as your main listening area or a different room.  I am also unclear if you need to be streaming to your main speakers and the remote speakers simultaneously.  If so, that is a bit tricky, because I don't know if a computer or streamer that will send a "conventional" digital signal to a dac can simultaneously send the same signal via Bluetooth or Airplay.

  If the remote speakers are in a different room, the old school  solution would be use an AVR that can do multiple zones in place of the Parasound, and then run speaker wires to those speakers, but you presumably have vetoed that.

   One solution for you might be to use Apple Home Pods as your remote speakers, and an Apple Computer as the streamer, and see if you can simultaneously stream and do Air Play.  It might work better if it stays in an all Apple environment, but I really don't know that, just speculating

I live in a three-story house.  The audio system, which is where the streamer would reside, is in the basement.  The remote speakers would mostly be on the first floor, one floor above the basement, or possibly outside, on my deck.  Both of these locations are well within strong signal range for Wi-Fi, but I am not sure how well it would work via Blue Tooth.

A wired solution won't work - I am hoping to have some sort of wireless solution.  Fortunately, I don't care if the wireless sound is audiophile quality.

Any thoughts on which solution - Bluesound, Sonos, etc., would be better?  I do not need to stream to both main speakers and remote speakers simultaneously.

What I want is to be able to "call up" music to be played on the remote speakers from an iPad or iPhone from the first floor without having to go downstairs into the basement to fiddle with the streamer.