Why do we treat time as parameter in non relativistic QM instead of treating position as parameter?
What is actually a parameter?
I have read this Wikipedia page, but could not understand why should we take only time as parameter?
A parameter is a variable you need to fix before you can apply a formula.
If you investigate a particular system on your table, you cannot even tell in principle in which state the system is unless you specify the time. (Except if the system is stationary, of course.) This makes time a parameter in classical mechanics and in quantum mechanics.
In classical (and quantum) field theory, you also need to point to a position before the event defining a state unambiguaously is determined; hence both time and position are parameters.
In classical relativistic mechanics of a single particle, you have the alternative to use eigentime as parameter; then time becomes a state variable rather than a parameter.
But there is no consistent definition of particle eigentimes for relativistic multiparticle systems. This is why relativistic multiparticle systems are always treated in an approximate way, typically the post-Newton approximation. An exact treatment would require a field theory, namely general relativity.
In less foundational contexts you may have many parameters. For example, you may have a function that depends on several variables but you keep some of them fixed. Then the variables regarded as fixed are termed parameters, as fixing them defines the instance of interest.
In Physics we observe things as functions of time. In Classical physics it is positions and velocities of a point-like bodies. In QM and other branches of Physics, it is waves or wave packets. Time is always an independent evolution parameter. The others are its functions, as we can figure out from experience.