Acceptable parametrizations of curved spacetime When we parametrize an (say absolute) space we have really no restriction (other than smoothness and one value naming) on parametrization. 
But I wonder on curved spacetime (also an absolute thing) there most be some restrictive condition of a acceptable parametrization. 
For example may be we shall consider an observable and parametrize time of events by his clock, or maybe some other subtleties shall be considered.
Is it true or we are free as much as spaces in parametrizing spacetime?
If we do so then there wouldn't be any subtleties in relation between parametrization time of events and clock of a free falling body?
 A: Absolute space is a Newtonian concept. Curved space a concept that comes from GR. They're both modelled as smooth manifolds. And as smooth manifolds they have lots of parametrisations. But I don't think that you're asking about this. 
Judging from the latter part of the question ('by his time') it seems that you're concerned about parametrising motion by proper time. This is a physically natural parametrisation but you can reparametrise the motion or worldline by some other smooth function but it's interpretation will no longer be physically natural. 
A: Indeed, there doesn't have to be any relation whatsoever between the $t$ coordinate and the time measured by any observer, or between the $x^i$ coordinates and anybody's rulers. There's often a tradeoff between convenience and interpretability. 
For example, in Schwarzschild coordinates, the time coordinate is indeed the time measured by somebody far away from the black hole, while the radial coordinate $r$ really measures the 'effective radius' of a sphere as given by its surface area. On the other hand, I'm not aware of any interpretation of Kruskal coordinates that's this nice, but they make the global structure of the spacetime much easier to see. To do this, you just need a coordinate system where light rays behave nicely, and light rays are not observers.
