Is it permissible to open an account to spread Islam on social media without the parents' permission, knowing that they object to it, and would he be sinful for doing so, and would that affect the expected reward?

Spreading Islam and inviting people to it through social media is commendable if the person spreading it is qualified for dawah. If the parents object to this, it might be due to their concern about the child's lack of qualification, their fear of unregulated relationships, or their apprehension about fitna (strife/temptation). In such a case, obedience to them is obligatory, in accordance with the verse: (O YOU WHO HAVE BELIEVED, PROTECT YOURSELVES AND YOUR FAMILIES FROM A FIRE) and the saying of the Prophet (peace be upon him): (EACH OF YOU IS A SHEPHERD AND EACH OF YOU IS RESPONSIBLE FOR HIS FLOCK). Therefore, one should ask the parents about the reason for their refusal, and it can be suggested that the account be under their supervision and participation.