Is it harmful to the Buddha when we request he or she to continue teaching instead of entering parinirvana? One of the ten things that Bodhisattva Samathabhadra says that a person on the Bodhisattva path should do is to request the Buddhas to continuing teaching instead of entering parinirvana. Given that a Buddha is fully compassionate and omniscient, he or she would want to teach if doing so is beneficial to sentient beings and would also be aware whether teaching is indeed beneficial to sentient beings. For example, the historical Shakayumni Buddha wanted to enter parinirvana because some of his sangha was becoming complaent that they would always have  the Thus Come One as their teacher. As such, it would seem that a Buddha would be able to decide well the timing of his parinirvana.

Given the above, I wonder how would it is appropriate for one learning to walk on the Bodhisattva path to request the Buddhas to continuing teaching. Is such an action for the benefit of the practioner to generate a wish of englightment and to take refuge in the Buddha? Does a Buddha need to enter parinirvana at a certain time, otherwise it would become harmful to him or herself or to the community at large? Is a Buddha unable to stay longer to teach if practitioners do not make requests? (Sorry that I am unable to phrase a precise question, but I hope I conveyed my gist and my confusion clearly enough.)
As I pointed out in my answer to "are earthworms our mothers?"  these kind of questions often assume incorrect frame of reference. The reason we request Buddha (or our local teacher as Buddha's substitute) to continue teaching instead of entering parinirvana, is not because it can literally make Buddha (or teacher) change their mind and stay. The idea is to create a mental context in which A) the teacher feels motivated to continue teaching because he sees the teaching is in demand, B) the students stay aware of the fact that existence of the teaching in this world depends solely on Buddha and then generations of teachers passing it to one another, and now to the students, and C) the student clearly realizes that the teacher could very well leave this place but is instead making effort to stay and teach solely out of compassion for the students. When this context is established the teaching can happen effectively, because both the sides will engage actively and with no resistance.

Granted these are entry-level practices, but they are very important to establish as foundation. Otherwise what happens if you skip straight to advanced teaching, the student may get as arrogant as to metaphorically grab the teacher by the hair and swing around. Needless to say, it hurts the student first and foremost.