I've looked it up in plenty of places like the Wikipedia page and such, and it is clear that the most common cause of Paresthesia is either a fair amount of pressure on a specific patch of skin causing lack of blood flow to the specific nerve endings in that limb (not to be confused with a stop in blood flow to the limb altogether) or a much stronger amount of pressure on that patch of skin for a shorter amount of time. This, although the most common cause, is not nearly the only one, which could be anywhere from simply sleeping on the wrong side of the bed to a lethal injection. I'm wondering what happens to the nerve cells that are affected at a cellular level, and what causes it at a cellular level. The level at which pounds per square inch aren't what is being noticed.
Underneath the superficial layers of your skin there are receptors which sense pressure, temperature and pain. These receptors are part of the peripheral nervous system which senses stimuli and they take the message conveying details about the stimulus to the somatosensory cortex of the brain. Here is where the perception of pain, burning, pressure etc is ultimately made. To take the simplest example, if you stop blood flow for a short amount of time in a limb, these receptors are activated, and will send signals to the brain that are interpreted as tingling or numbness. With more severe pain, different receptors are activated which , again, project to the same brain area but a different message is read out. If the pressure from one limb is removed, the receptors will go back to normal function as blood flow is restored.