**Core Concept**
Peripheral neuropathy due to vitamin deficiency primarily affects the distal nerves, leading to damage of the myelin sheath or the axons. This condition often results in weakness, numbness, and pain in the affected limbs.

**Why the Correct Answer is Right**
The correct vitamin deficiency associated with peripheral neuropathy is vitamin B12 (cobalamin) deficiency. This is because vitamin B12 plays a crucial role in the synthesis of myelin, the fatty substance that surrounds and protects nerve fibers. A deficiency in vitamin B12 can lead to demyelination, which disrupts the normal functioning of the nerves, resulting in peripheral neuropathy. The enzyme methylmalonyl-CoA mutase, which is involved in the metabolism of fatty acids, is also dependent on vitamin B12 as a cofactor.

**Why Each Wrong Option is Incorrect**
**Option A:** Vitamin E deficiency is associated with neurological symptoms, including ataxia and tremors, but it is not the primary cause of peripheral neuropathy.
**Option B:** Vitamin D deficiency is primarily associated with bone health and calcium metabolism, and is not directly linked to peripheral neuropathy.
**Option C:** Thiamine (vitamin B1) deficiency is associated with Wernicke-Korsakoff syndrome, which affects the brain, but not typically with peripheral neuropathy.

**Clinical Pearl / High-Yield Fact**
Peripheral neuropathy due to vitamin B12 deficiency can be prevented by consuming foods rich in vitamin B12, such as meat, fish, and dairy products, and by taking supplements if necessary.

**Correct Answer:** C.