Does a foreign parent that sets up a US subsidiary to process global payments to non US customers have to pay US tax?

We are a foreign company. We sell online software. We'd like to set up a U.S. subsidiary 1) to accept payments globally & 2) to sell to U.S. customers. All sales to non-U.S. customers would be done by the foreign parent, but payments would be processed by the U.S. sub. All sales to U.S. customers would be done by the U.S. sub. Obviously, all income from sales by the U.S. sub to U.S. customers would be taxed by the U.S. But, would income from sales to non U.S. customers BY the parent that are processed by the U.S. sub be taxed by the U.S.? In other words, if payments for non-U.S. sales BY the parent went through the U.S. sub's payments platform, then into the U.S. sub's bank account, and then were transferred to the parent's foreign bank account, would those funds be taxed by the U.S. before they are transferred to the foreign bank account? We are referring only to sales by the foreign parent to non-U.S. customers where the payments are processed by the U.S. subsidiary.