Is it permissible to supplicate for one's parents via the internet, and what is the ruling on spreading unauthentic supplications that people compose and circulate as if they were established legal supplications, and does Sheikh Muhammad Al-Munajjid's fatwa apply to them?

Shaykh Al-Munajjid did not warn against the aforementioned supplication; rather, he prohibited the generalization of adhkar (remembrances) that are man-made and chosen by individuals, out of fear that they might be taken as legislation for which there is no textual basis. There is no harm in supplicating for parents online or elsewhere, with any good supplication, although it is better to supplicate with the ma'thur (transmitted) supplications.