No. The Actions menu at the top-right in the parent project's details pertains to the parent project only. So, if you select the phases on the Structure tab of the parent project and then send to QuickBooks Online using the Actions menu > Send To, BQE CORE sends the parent project only, not the phases. This is because in CORE, a phase is an independent project record. You need to go to the detailed view of the phases to send them to QuickBooks Online. You can also send multiple phases in a batch or all project records from the Settings > Integrations screen by fine-tuning the sync settings for QuickBooks Online integration. Check CORE Help Center for details.
CORE Help Center