what's the difference between "ringing bell" & "bell ringing " here:
1- I was awaken by a bell ringing.
2- I was awaken by a ringing bell.
and "car parked" & "parked car" here:
1-there was a car parked outside
2-there was a parked car outside
1- I was awakened by a bell ringing.
2- I was awakened by a ringing bell.
The difference is subtle and contextual. "A bell ringing" is really talking about the bell. You woke up and heard a bell, which you realized was ringing, as opposed to gonging or chiming, I suppose. Or you woke up and heard a ringing, which you realized was coming from a bell, rather than a phone or something else. There isn't really much difference between the two without knowing what comes after this scene, or what has come before. If this is part of a larger story, for example, either the bell or the ringing could be important to the characters, and making the distinction is important.
The same is true with "car parked" and "parked car." Their order tells us which part is more important. Do we need to know that it's a car that's parked outside, or do we need to know that the vehicle outside is parked, in which case the fact that it's a car is secondary.
Maybe there's more to this on a grammatical level, and someone else could explain the mechanics of this more precisely, but as a fiction writer, this is how I see and use the difference.
There is another difference in meaning that the other answer didn't provide.
a bell ringing
This could be interpreted as an event: "We're going to have a real bell ringing." In other words, one or more people are actually involved in the act of ringing a bell.
a ringing bell
This simply means that there is a bell that is ringing. It may or may not be the case that somebody is ringing it. Potentially, there could be a device inside the bell that is causing it to ring—or it could be that it's designed in such a way that it's already been rung but the sound is continuing for some time afterwards.
A similar analysis could be given for a car parked (verb) and a parked car (noun).
These sentences have very similar meanings. The first sentence is saying that the ringing sound woke you up, and the second sentence is saying that the bell woke you up. Usually, these have the same meaning.
However, suppose that a bell is ringing quietly, so that it doesn't wake you up, but then the bell falls on your head while it is still ringing, and the blow to your head wakes you up. Then it would be accurate (but strange) to say "I was awakened by a ringing bell", but it would not be accurate to say "I was awakened by a bell ringing."
The first sentence is saying one thing about the car: it's saying that the car is parked outside.
The second sentence is saying two things about the car: it's saying that the car is parked, and that it is outside.
In practice, these two sentences mean exactly the same thing. However, the first sentence is the common way to say this; the second sentence sounds a little strange.