The dark areas are "unaged" as they have not been bleached out by the UV rays from the sun. Others on here will give you a better idea of what you can do to attempt to get the colors the same.

I have only come across this issue once. A floor had area rugs on them so long that the light had bleached the perimeter of the room. Just like yours, it did not sand out. The only solution I can suggest is to find a light/medium stain that would be close to the color of the centers of the rooms and stain the entire floor. IMO, trying to stain the areas different isn't likely to work. For the best result, apply a light coat of water to the entire floor, let it dry then apply the stain. You can add up to 8oz. of the same stain to each gallon of oil-modified polyurethane and apply two coats of tinted poly, abrading lightly between coats. Abrade the second coat and apply a final coat of clear satin oil poly.
It would be ideal if you have an area where you can make a sample first.
This process should make the center of the room slightly darker and the edges of the room much darker, and address the issue of the Pine possibly being splotchy from the staining process.

Pick a stain that is the color of the middle tones of the darker area and apply it over the entire floor.

If that doesn't even it up enough, find some boards that cross the light/dark boundary and stain them a couple shades darker than the first stain to break up the lines of the dark patch. Pick random boards -

Also darken a few boards that are completely inside and outside the dark patch to make it look like natural color variation between boards.

Use a pre-stain conditioner to prevent "grabbing" and blotching.

We had something similar in our house. The parts that were worn when we removed the carpet as in this photo...

Were the parts that turned out darkest when we sanded, sealed, and poly-ed.

I guess the shellac/lacquer that was on there protected the wood from aging. Interesting.... We opted to just leave it as-is and call it character. :-)