Horse manure tends to be weedy just by definition :-) Horses consume a lot of weedy plants and their digestive system doesn't process them as thoroughly as say cows with their multiple stomachs do. Hence, horse poop tends to generate a lot of weeds.

About the only way to control them is to get the pile to heat up to seed killing temperatures. Very likely you will need to add other organic matter to make this happen. Horse manure often contains a lot of bedding and this slows the process and limits the heat the manure should generate, Adding some "greens" - nitrogen sources like grass clippings, plant trimmings or chicken manure, blood meal or alfalfa meal will help.

And aerate the pile by turning periodically if you can......your tractor should be helpful for this :-)

If you can raise the temperature of your pile up to at least 140F (160F is even better) and hold it there for 3-4 days, the bulk of the weed seeds should be killed off.

Animal manures left to sit around and age can grow many "weeds" both because some of those seeds may not be killed during the digestion process of the animal, However, seeds from unwanted plants also are put there by the birds and wind and those will germinate and grow quite well, not all of the unwanted plants in manure piles were there while the animal ate the hay.
Proper composting of that manure, not simply allowing it to sit and loose nutrients, is the best way to control unwanted plant growth.

You could cover those new beds with several sheets of newspaper after planting them. Then put a weed free mulch on top of the newspaper. Or use cardboard. Something to block the weeds from growing through.

Vermiculture might help, use the manure to feed worm beds then use the worm castings for your flowers. Having a source for plenty of horse leavings is a big step up in the process. Have a look at medopetes carpet beds in the vermicomposting forum.

Ooops, just saw zone 5. This would make compost worms more difficult I would think depending on how harsh winters are and what you can do to protect them during the cold.