Is it permissible for an employee who lives with his parents and uses public transportation at his own expense to receive a transportation allowance designated for living in his distant apartment, knowing that the company provides transportation in the area where his parents live, but he does not use it?

If what you are doing is a trick and deception against the company, such as living in a place where company-provided transportation is available, then it is forbidden (haram). However, if the company does not care about this and grants you transportation allowance regardless of your place of residence, then there is no blame upon you. The main point is to refer back to the company and inform them of the matter; if they agree to grant you the allowance, then there is no harm. If they prohibit that and provide you with transportation, then the matter is likewise. The essence lies in the agreed-upon terms of the contract, based on the Almighty's saying: ﴿O YOU WHO HAVE BELIEVED, FULFILL [ALL] CONTRACTS﴾ and the Prophet's (peace be upon him) saying: «Muslims are bound by their conditions.»