Most properties can be animated by right clicking and selecting 'insert keyframe' but the keyed particle key properties (properties such as 'Time:' and 'Duration:') don't give that option when right-clicked. Right clicking the box containing the list of particle keys does give the option, but adding a keyframe for that doesn't seem to do anything. Is there any way to alter these particle key properties?
Blender lets us animate "almost" any value that we can adjust, there are some values that do not support being animated. These either don't make sense to animate or might cause issues if they are animated. Thse values will not have any keyframe options when right clicking on them, if you press I with the mouse over them you will get a warning message at the top of the window telling you it cannot be animated.
The number of particles, start and end of the particle emission are some of the values that we cannot animate. You will find others, like initial orientation and velocity, as well as physics type.
Some other values like render resolution and start/end frame also cannot be animated.