The attacks follow Friday's Saudi raids on the important port city of Hodeidah in Yemen.
The Iran-backed Houthi movement says in a video statement that the attacks are aimed at the Saudi oil company Aramco, which has facilities in Jizan and Yanbu on the Red Sea.
"We are expanding our operations and will escalate (the attacks) in the coming days," military spokesman Yahya Saree said in the statement.
This is escalating the conflict in the Red Sea - the fighting has a direct bearing on the ongoing war in Yemen and the conflict in the Middle East.
The Houthi movement - which de facto controls large parts of Yemen - announced on Friday that several telecommunications and communication facilities had been hit in attacks on the province of Hodeidah (al-Hudayda) in Yemen and explosions had been heard. Attacks were also said to have targeted the island of Kamaran.
The Houthi movement warned Saudi Arabia after the attack, calling it a "serious mistake that will cost the Saudi regime dearly," in a statement.
In addition to the attacks on oil facilities, with drones and missiles, a Saudi ship was reportedly hit in the Red Sea on Friday evening - the damage is said to be limited.
The Houthi movement claimed earlier this week that it has initiated a naval blockade in the Red Sea, targeting Saudi Arabia.
