Is it permissible to abandon one's father and sever ties with him due to his mockery of religion, the Prophet (peace be upon him), and the Companions, and his ridicule of Islamic studies students and bearded individuals, in addition to his causing psychological harm to his son?

The father's right is great and is not forfeited by his committing sins. It is not permissible to abandon him or shun him due to his sin. Allah has commanded companionship with kindness towards a polytheist parent, just as honoring parents is obligatory even if they are disobedient or polytheists. One must strive to guide him with gentleness and wisdom, seek help from siblings, diversify methods of sincere advice and dialogue, or seek help from someone who influences him, or let him listen to audios of scholars, while treating him well and honoring him to gain his trust.