Is it permissible to sell or market a project that I do not own, but with the approval of the parent company, and is it permissible to take a commission for this if the transaction is completed with a reduction in the project's value, with the knowledge of all parties?

It is permissible for you to be an agent for the parent company to sell the project to another party for a fixed sum or a percentage. This falls under the category of permissible brokerage, and in this case, your role is limited to marketing and selling. It is also permissible for you to contract with the parent company to execute the project, and then subcontract with another contractor to execute it for a lesser amount, with the difference being yours. However, you will be responsible for the project to the parent company until it is completed and delivered according to specifications and conditions, unless it is stipulated that you must perform the work yourself.