One aspect that the other answers do not take up is performance.

If you are programming in a sufficiently low level language (C, C++, assembly) a large number of parameters can be quite detrimental to performance on some architectures, especially if the function is called a large amount of times.

When a function call is made in ARM for instance, the first four arguments are placed in registers r0 to r3 and remaining arguments have to be pushed onto the stack. Keeping the number of arguments below five can make quite a difference for critical functions.

For functions that are called extremely often, even the fact that the program has to set up the arguments before each call can affect performance (r0 to r3 may be overwritten by the called function and will have to be replaced before the next call) so in that regard zero arguments are best.

One aspect that the other answers do not take up is performance.

If you are programming in a sufficiently low level language (C, C++, assembly) a large number of parameters can be quite detrimental to performance on some architectures, especially if the function is called a large amount of times.

When a function call is made in ARM for instance, the first four arguments are placed in registers r0 to r3 and remaining arguments have to be pushed onto the stack. Keeping the number of arguments below five can make quite a difference for critical functions.

For functions that are called extremely often, even the fact that the program has to set up the arguments before each call can affect performance (r0 to r3 may be overwritten by the called function and will have to be replaced before the next call) so in that regard zero arguments are best.

Update:

KjMag brings up the interesting topic of inlining. Inlining will in some ways mitigate this since it will allow the compiler to perform the same optimizations that you would be able to do if writing in pure assembly. In other words, the compiler can see which parameters and variables are used by the called function and can optimize register usage so that stack read/write is minimized.

There are some problems with inlining though.

1. Inlining causes the compiled binary to grow since the same code is duplicated in binary form if it is called from multiple places. This is detrimental when it comes to I-cache usage.
2. Compilers usually only allow inlining up to a certain level (3 steps IIRC?). Imagine calling an inlined function from an inlined function from an inlined function. Binary growth would explode if inline was treated as mandatory in all cases.
3. There are plenty of compilers that will either completely ignore inline or actually give you errors when they encounter it.