You’re right. The picture in the OP looks a lot like the persimmons I’ve seen growing in the wild, less like the ones I see at the grocery store, but they definitely don’t have those things growing on them in the wild.
Sorry - my misunderstanding.
Loquats have quite distinctive large, thick corrugated dark green leaves, also, they typically flower in late autumn and fruit in early spring.
A cross-section of the fruit should help (although hawthorn fruits do look almost exactly like tiny apples, even on the inside).
Most hawthorn species have conspicuous thorns on the twigs; crab apple trees can sometimes be a bit wiry-prickly, but are not usually thorny.
I’m puzzled by earlier references to grapes though - do you mean that the fruit is squashy like a grape?
Some of the ornamental apple species do have fruits that are a bit translucent.
The fruit on the lower left in your photo does show faint signs of longitudinal patterning that is typical of apples.
The only thing that doesn’t seem quite right to me about this being an apple variety is the green-ness of the calyx, but these may not be fully-developed fruits.
Could it be some type of plum?
What are the leaves like? If the leaves are long and narrow, I’d definitely vote for a loquat tree. The fruit look very much like loquats and the shape of the tree is loquatish, but the leaves would seal the deal.
They do have that growth on the endthat Colibri mentioned.
Probably not, but maybe a large variety of Rosehip? Popular in Denmark apparently. Same shape, but maybe not waxy-rinded enough.
Who are you, and what have you done with Biggirl? :dubious:
No. Plums do not have a persistent calyx
Loquats have a distinctive specled patterning on the skin, not evident in the OPs picture.
Wow, really? I’ve always eaten kumquats raw/fresh, including the skin.
Apparently some types can be eaten raw, but not all: Kumquat - Wikipedia
Where I live they are used to make a sweet liqueur which I hate so I am irrationally prejudiced against kumquats. 
C’mon, Biggirl. Go back to the park. Take a picture of the leaf. Put us out of our misery!
This afternoon I told my husband we’ve gotta go back to the park. Valley Stream State Park is tiny but cute. Plan to bbq there next week in the middle of the week during the day when no one is there.
Excellent. Nobody to photobomb your leaf pics 
Where is Mangetout when you need him?! Oh, he’s right upthread.
I’m pretty sure it’s not a loquat. The leaves in picture linked #1 were too abundant and small. And the shape of the fruit was right, but I think they were too orange/red to be loquats. As noted above, loquats are blotchy and they’re more yellow-skinned.
–G!
Tomorrow I’m going back to the park. I’ll get a pic of the leaves and the inside of the fruit. If it’s still fruiting.
I don’t think drupes get that growth on the end. That rules out peaches and plums.
Googling images of crab apple fruits, I’m convinced that this is what Biggirl’s mystery tree is. It’s just a matter of finding the particular variety. The John Downie variety mentioned upthread is certainly at strong contender.
Incidentally, since haws were also mentioned, I thought I’d say that a rose expert once told me that hawthorns are thought to be the ancestors of the modern rose. Just thought I’d throw that out there for no particular reason.