Is it permissible to establish a private software company and sell programs not produced by the parent company, provided that clients are not taken from the parent company and private company work is not conducted during official working hours? Is it permissible to deal with a client who was previously offered to the parent company and rejected its offer? Is it permissible to accept a client from the parent company's clients if they come to us of their own accord?

There is no sin upon you both for establishing a company that sells the same software as your current company, provided that your work is outside official working hours, and that you do not exploit your work to promote your new company or divert clients from the company you work for. There is no sin in dealing with clients who come on their own accord. There is no objection to contacting the person who refused to complete the deal with your company and offering him your products.