Is time not a universal parameter, really, in special relativity? I want to check that I get it right.
The way that I have it in my mind is that, in my frame of reference and from my point of view, SR associates only one time parameter for the whole space, and not distinct one for each point in space.
And that it's only when we want to translate the view of the world from a frame of reference to another, do we need to use Lorentz transformations which mix space with time, depending upon the usual parameters ($v$, $x$ and $t$).
And that when we say moving clocks evolve more slowly, we don't mean that time depends on the velocity of that clock, because in that case we would be associating time with objects and not with the points of space itself. From our frame of reference, the point at which the clock exists has a time parameter equal to all other points in my frame.
Am I wrong/correct?
 A: You're mixing up two things in this question:
1) How we label individual points (Einstein calls them "events") in space-time
2) What results someone would get when looking at their clock when they pass through spacetime points.
The first thing is almost completely arbitrary, especially in full general relativity.  The second thing is an unambiguous result of an experiment.  What Lorentz transformations do is mix together spatial coordinates and temporal ones, so one observer's space coordinate looks like it partially depends on another observer's time coordinate.  What all observers will agree on, though, is how much time will pass on observer A's clock as A moves from spacetime event $p_{1}$ to spacetime event $p_{2}$
A: Yes, really.
But that doesn't mean that we're floating around loose without a foundation.
In Galilean relativity all observers could agree on a number of things about a interaction of process. Things like


*

*How long it took (the same for all the bits)

*How much the distances between objects changed in that time

*What the mass of each bit was

*...


In Einsteinian relativity all observers can still agree on a number of facts, they are just different facts:


*

*The proper time experiences by each bit (not necessarily the same anymore, but uniquely defined)

*The interval of the whole interaction.

*The mass of all the bits.

*...


The point is that we can still do physics. You just have to give up this notion that time and space are independent of one another.
A: 
I want to check that I get it right. [...]
  [...] we would be associating time with objects and not with the points of space itself. 

In physics, the word "time" is used in various related but quite different meanings (which I describe in some detail below). 
Therefore, if you care to "get it right" (which is of course commendable) you should avoid using the word "time" altogether in favor of established terminology which is distinct and specific to the various meanings which can and should be distinguished:
(1a): "time" meaning one particular "indication" of one particular participant; in accordance to Einstein's prescription
"[... that instead] of ''time'' we substitute ''the position of the little hand of my watch''." [Punctuation marks as in the German original: Ann. Phys. 17, 891 (1905)],
(1b): "time" meaning an ordered set of some, or of all, indications of one particular participant;
e.g. in the phrases "the time when the dinosaurs roamed the Earth", or "the time while I was on vacation" etc.,
(2): "time" (and especially "proper time, $\tau$") meaning the "duration" of one particular participant, from one particular indication until another;
e.g. the duration of life of some specific unstable particle (which is often less distinctively referred to as "proper lifetime"), or "the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of [...]",
(3): "time" (and especially "clock time, $t$") meaning some parametrization of indications (in terms of real numbers), i.e. of an ordered set of some, or of all, indications of one particular participant, 
(4): "time" (and especially "coordinate time, $t$") meaning some parametrization of events (in terms of real numbers), i.e. of some ordered set of (non-intersecting) space-like hypersurfaces (or suitable discontinuous generalizations) of events.

SR associates only one time parameter for the whole space, and not distinct one for each point in space.

Well: SR, and physics and geometry in general, are independent of any particular parametrization, or assignment of coordinate (tuple) values.  

And that when we say moving clocks evolve more slowly [...]

"We" (physicists) would not reasonably say somesuch.
Instead, in keeping with the theory of relativity, we can reasonably say that
"motion, and geometric and kinematic relations between participants in general, must be taken into account in order to compare durations, and to compare (proper) rates". 
A: No.
My answer is negative, even if I confirm the statements of other answers: 

"The first thing is almost completely arbitrary, especially in full
  general relativity. The second thing is an unambiguous result of an
  experiment."(Jerry Schirmer)
"In Einsteinian relativity all observers can still agree on a number
  of facts, they are just different facts: The proper time experiences
  by each bit" (Dmckee)

It is a fact that any observed time is relative, observer-dependent. And it is a fact that the only thing we know for sure about time is the proper time of each object (Just try to find out a counter-example, there is none)
By consequence, if the time of a clock A is measured by the clock of an observer, we know that this observed time is in contradiction with other observers. Only the clock A tells us the time which is valid for all observers: Its proper time.
By consequence, time is not an absolute parameter, it belongs to each object. And spacetime is an observer's view which is universal (concerning the whole spacetime), but which is relative (no absolute value) and volatile.
