Are there any political parties in Saudi Arabia? If there aren't any, then what stops the establishment of such parties?
Are you asking about officially recognized parties, or actually existing (with or without recognition?)
Wikipedia has a list of Saudi political parties that exist in reality.
However, none of those parties are officially recognized under the Kingdom's law, and are officially prohibited (reference: Marshall Cavendish. "World and Its Peoples: the Arabian Peninsula" (2007). page 78)
For a more primary reference, this can be seen in Saudi The Basic Law Of Governance, issued in 1992 by The Custodian of the Two Holy Mosques, King Fahd Bin Abdulaziz Al-Saud (Source: website of Royal Embassy of Saudi Arabia in Washington, DC):
CHAPTER THREE: THE VALUES OF SAUDI SOCIETY
Consolidation of the national unity is a duty. The State shall forbid all activities that may lead to division, disorder and partition.
Mass media and all other vehicles of expression shall employ civil and polite language, contribute towards the education of the nation and strengthen unity. It is prohibited to commit acts leading to disorder and division, affecting the security of the state and its public relations, or undermining human dignity and rights. Details shall be specified in the Law.
Saudi Arabia is an absolute monarchy. What stops the establishment of political parties is said absolute monarchy.
No political parties or national elections are permitted