Is the sentence badly phrased?

Could somebody please help? Thanks.

Sure. I'm trying to figure out where to start. First, the part about falling down. I'm guessing that a car park is what is called a parking lot in the U.S.
So it's really not a road. "Fell down" is a common expression, but "down" is
assumed. I think "...I tripped and fell at the car park." is sufficient.
The part about resuming work during a hospitalization leave leaves me baffled. Someone is either working or on leave. So I don't know what that means.
I suggest you break it into two or more sentences, first telling about the fall and hospitalization, then resuming work.
How about "In February of 2016, I tripped and fell at a car park, and I was hospitalized as a result. Shortly after, I resumed work "

Many thanks, Ponderer.

The sentence is not mine. I extracted it from a magazine.

Perhaps I should apply for a job as editor?

Before I returned to work after my hospital stay, I tripped and fell down. I was on the road in a car park when I fell down.

you could have fell sideways not necessarily down. down contributes to clarity on direction of fall

hospital stay occurred before fall not as a result of the fall. keep your present job

Thanks, bunnyhabit, for your explanation.

May I know whether the original sentence is ambiguous?

Evidently

Thanks, Ponderer.