Is it time or duration? Taking this post: "Is there a proof of existence of time?", as a starting point. Therein was mentioned that there is confusion between:
"time" and "flow of time".
There was a comment (of mine) that the confusion is not between time and flow of time (which are equivalent), but between time and duration of which one is a dimension (i.e duration).
Given the importance of the problem of time in General Relativity and Quantum Gravity.
Having made this disctinction is an important step, since duration can easily be considered as a dimension (with the proper $c$ factor) along with other space dimensions, than actual time (or flow of time).

Can we say that time parameter/dimension in SR/GR actually
  represents not event time but duration (i.e time-interval)?

By the way this would elucidate the wick-rotation method, as transforming from "duration" to "frequency" representation.
(Not to mention that one can have as many duration dimensions in a manifold as one wants with no conceptual or definition problems like when one attempts that with extra time dimensions, per some theoretical proposals)
 A: Duration is certainly a more physical concept than time. 
Duration is something you may measure between timelike separated events while time is always something you compute by adding up duration measurements + an arbitrary constant to fix the origin. 
Duration is experimental and relational while time (e.g. GPS time) is an abstract a posteriori construction. 
For these reasons I think your proposal is correct.
It could happen that advanced theories get rid of most of Time, but at some point these advanced theories will need to have some room for durations (even if only in a limit).
Beyond this quantitative component of Time, or rather more fundamentally, there is also the more qualitative notion of ordering of timelike separated events, linked to causality. The ordering does not require a continuous flow, e.g. discontinuous "pre-time" can work for this purpose. In particular if durations are discrete in a way or another. This would be quite a different concept for time.
@BrandonEnRight : in such a domain as modern gravitation theory, it is normal to have discussions on fundamental concepts because they need to be questioned and understood to see better what is useful in the postulates of the theory under construction. I understand you want to contain the pseudo-scientific spontaneous trends of discussion, but this not the case here. And about reverting to philosophy SE, I would say yes if we were into metaphysics (discussion on non-experimentable concepts). But here we are still in physics as all the assertions lend themselves to experimental tests, at least in principle
A: Okay, I am going to try and give this a shot, but this is most probably not going to be a decisive answer.
Let us operate with the term event time and duration and consider only special relativity (SR). The conclusions of general relativity should be the same for reasonable space-times. (e.g. without closed time-like curves etc.)

We expect event time to identify the exact relation to all other events, i.e. at a given event time, we choose for events with a smaller time to be in the past and the ones with a larger time to be in the future. Notice that this notion of time does not really need any quantitative measure, it is more a question of a certain topological ordering of events.
However, the famous Rietdijk-Putnam argument shows, using relativity of simultaneity, that there is no natural global classification of such an order of events (at least without invoking a privileged class of observers). In special relativity, there is thus no global notion of event time. 
It would seem that event time would make sense in the lightcone but it is not so. We can always order causally connected events, but for non-zero time-like intervals between them, there will always be adjacent events in their lightcones that are causally disconnected. I.e. there is never a nonzero volume of space with unique event time. The only case of event time in SR as I define it is the proper time on a single time-like curve and it's homeomorphisms.

As for duration, we want it to measure an amount of time passed. But how do we do it? We must compare the amount of time to a certain physical process. Galileo used his heartbeat, but we would use the cycles of radiation in the caesium atoms. In this sense, the physical process must always be happening in a certain frame of reference and special relativity tells us (and this is verified by experiment e.g. through mean decays of particles) that the duration of any physical process is happening with a stable rate with respect to the proper time in a given frame of reference.
Different observers will thus through time dilation have different notions of duration of processes observed in their surroundings. It is pretty easy to show that once again not even the ordering of magnitude of duration of physical processes is universal. As in the case of event time, there is no natural global definition of duration and a natural fixed comparison of durations is only possible on the world-line of the observer but nowhere else.

To conclude, there is thus actually no unambiguous time dimension/coordinate in relativity, be it event time or duration. You need four numbers to specify your event, but none of these are uniquely identifiable with either space or time (without a mixed-in part of the other one) unless you specify the observer who is asking that question.   
A: The answer is more simple than you think. Time is that, which is measured by (technologically suitable) clocks. Physical theories will simply tell you how clocks behave under certain conditions. This is purely descriptive. There is not a single physical theory out there, that gives a microscopic description of time, although the similarity of time with irreversible thermodynamic processes SUGGESTS, that it can be derived from a state counting argument in a microscopic theory of spacetime using methods from statistical mechanics. Such a suggestion is far from being a useful theoretical framework, of course. 
