Between a parent company and a subsidiary in Dubai, are corporate transactions (sale of good/services) subject to taxes? If yes, what types of taxes?

Yes, these Transactions are liable for Tax Liability as per the provisions of the Country of both Principal and Subsidary

1.  The Export duties and other allied fees would have to be borne by the principal and at the subsidary the required Import duty alongwith other allied charged has to be paid by the subsidary. In this scenario, wherein the Principle Company is located in Dubai, the export duty is exempted and the parent company would only have to pay for the transportation and other charges.