No, insects do not primarily breathe through their skin. Instead, they breathe through a system of tiny openings called spiracles located along their bodies.
Insect Respiratory System: Spiracles and Tracheae
Insects have a unique respiratory system unlike mammals or amphibians. Instead of lungs, they utilize a network of tubes called tracheae. These tracheae branch throughout the insect's body, delivering oxygen directly to the tissues. Oxygen enters the tracheal system through the spiracles, small openings found on the sides of their thorax and abdomen. Carbon dioxide, a waste product of respiration, exits the body through these same spiracles. [Source: Ask A Biologist, Integrated Crop Management, Six-Legged Science, Quora, r/educationalgifs]
While some insects may exhibit minimal gas exchange through their skin, this is not their primary method of respiration. Cutaneous respiration, or breathing through the skin, is common in amphibians and some other animals with moist, permeable skin. However, insects possess a hard exoskeleton which prevents significant gas exchange across their body surface. [Source: Cutaneous respiration - Wikipedia] The exoskeleton's primary function is protection and support, not gas exchange. [Source: Insect molting is 'like having your lungs ripped out']
Examples of insects utilizing spiracles for respiration include cockroaches, as explicitly stated in one of the provided references. [Source: Reference text provided].
The process of molting, where insects shed their exoskeleton, temporarily disrupts their respiratory function. This is because the spiracles are part of the exoskeleton, and their closure during molting prevents gas exchange. [Source: Insect molting is 'like having your lungs ripped out']