The idea of using a parachute is to let it drape downwards to help warm up and close in those massive conference and ballrooms.
If you center it in the room, you could attach fabric or paper streamers to the edges of the parachute ( only if this is just a decorative parachute- not one that is expected to function obviously) and drape them to the edges of the room.
Another idea would be to center the parachute over the dance floor. That will break up the ceiling so it doesn’t look so big. I think this is what I would do.
Your idea to have lights under it is excellent. It will be a focal point of your decor.
You said the ceiling had ceiling tiles. Are they the kind that are hung from a T bar framework? The advantage of that system is that it is easy to hang things from the bars that support the ceiling tiles. They even sell special clips for that purpose, or you can just lift up the ceiling tile enough to slip some fishing line over the bar.
You might want to consider hanging some lattice panels paralell to the ceiling. It would go with your floral, mid summer night’s dream concept. The panels come in various sizes, up to 4’x8′. You don’t have to cover the whole rest of the ceiling, just space them out evenly to block off some of the ceiling. You could put lights on these also, or also weave in some artificial vines
I do decorating on the cheap for charities so I am not above looking for ways to cut costs. I would place an ad on craigslist and see if anyone planning to build a fence using lattice, and would want to share the cost of the panels with you- 1/3 by you and 2/3 by them. It would cut costs for both of you.
Another idea is to hang large branches parallel to the ceiling. These can’t be little branches that you would use for a centerpiece because of the size of your room. I have obtained branches like this free- also off craigslist. You could also contact landscaping and arborist companies. They often chip them up on site, so you might have to be able to meet them at the job site.
If you find a souce for branches, you can also pot some up using plaster of paris to keep them upright and use them to make groves of trees for the corners of the room.
The challenge with using large branches is you need a place to store these things until the wedding.
PM me if I can be of any more help.