I agree - either seedlings or suckers. If they are seedlings they will pull out. If suckers they are attached to roots or trunk of the tree and will not pull. I'd also like to know what kind of tree they are under.

If they are tree seedlings, they can't survive repeated mowing.

It is under some type of bradford pear tree, last fall I didn't collect the fallen fruit in time, when I got to it, the fruit was rauten and soft to the touch when I picked it.
The picture is of bradford pear seedlings and I don't know if I should spray them with some 2, 4 D Broadleaf Weed Control or just keep mowing and let then die naturally.

Mow them asap and keep doing it.

If they are seedlings, mowing as often as you would normally to maintain your lawn should kill them. If the are suckers from the roots, you may need to trim them a bit more often, and lower to the ground, before they give up. If they are root suckers, poisoning them will poison the tree. The less the tree is pruned, the less it should feel like making root suckers.

Not sure your tree is a Bradford. The fruits on that are too small for anyone to think necessary to pick up.

They are seedlings and not suckers from the roots;
I can pull each one by hand, complete with its root system. The tree is Bradford
pear tree, my HOA had this tree type planted in the center of each front yard throughout
the neighborhood, and it does look good for one week a year when it blooms.
Looking in my neighbor's front yards confirmed this; they all have tree seedlings growing under the
trees, some neighbors try
to eradicate them with herbicide of some kind and ended up with bear spots
under the tree (Dead lawn).

I blame myself for the large number of
seedlings, last fall I left the fallen fruits on the ground and picked them up
only when I was racking the dead leaves, at the time the fruit under the leaves
was decomposing and very difficult to rack, I had to scoop it by hand, it was a
very messy job.

The pics and info here should help you compare your tree to Bradford:
http://webpage.pace.edu/naturespace/Bradford%20Pear.htm
http://www.invasive.org/browse/subinfo.cfm?sub=10957

It's possible but seems unlikely that an HOA would use these trees as
you describe. The flowers smell awful, and these trees are very
short-lived as far as trees go, about 30-40 yrs. Once they mature, they
start falling apart in large chunks. They are common in the strip of
grass between sidewalk and street, but unlikely as "the" tree in a front
yard. If that's what it is, I'd quit worrying about the sprouts &
investigate getting rid of the tree.

Some type of crabapple seems much more likely from the info so far.

I am sorry to say that matching the leaves, fruit, flowers,spring and fall colors, my tree is some kind of Breadford pear tree, the neighborhood is fifteen years old so none of the trees started to fall apart, the only picture that I have of my tree in fall colors from 2011.

Oh man. It does look like BP, no doubt they are very pretty. In a
well-manicured lawn like that, mowing should be all that's needed to be
rid of the seedlings.

If your HOA gets ready to make any more landscaping decisions, keep a close eye on them. Love your pretty house, BTW!

Our
neighbor across the street had this happen to his one day last summer,
an ordinary not-windy day. About 1/4 of the tree just fell off:

Hmm... pic not working, will try another time.

(Edited to add pic.)