Why do rubber parts like thorns stick out in new car tires?

You are talking about the cuts like in the picture above. These are called Vent Spew. Found on new tires. They are not created intentionally, nor do they enhance a tire's performance. They are created when a tire is cured.

Tire curing is a process in which green tires made of soft rubber are vulcanized using pressure and heat to harden the tire into its final shape.

Things that are needed while curing tires are:

Tire Mold:

It is a mold made of metal. It can be separated into two or more parts and the mold is brought back to its original state by inserting the green tire into it.

It is green tire. The components used in tire manufacturing such as bead, rubber ply, steel belt etc. are joined together to give a basic shape of tire. The green tire remains in the mold during curing. Green tires have soft rubber.

It is tire curing bladder. Made of rubber. During curing, the bladder is inserted into the green tire.

The tire curing process is done in a tire curing press. In this process, hot water or steam is introduced into the bladder and high pressure and heat is applied to the green tire, as a result, the green tire takes the shape of the mold, and it vulcanizes to the shape and strength of the finished tire.

During this process, some air is trapped between the green tire and the mold. And if this air is not removed, some holes are formed in the finished tire. Some fine holes are kept in the mold to release the air. These are called spew holes or vent spruces. During curing, due to the high pressure and heat, some of the rubber tries to escape through the spew hole with the air, so we see vent spew in the finished tire.