The new version of the Parent App, 1.0.8, adds settings to control on a per-alert and per-phone basis, whether to generate a user-notification when the alert is received.
The alerts will always be delivered to the Parent App, these settings just control whether your own phone will make a sound and vibrate when the alert is received.
So even if you decide to disable notifications, the alerts and reports will still be available for viewing on the Parent App (or the Parent Web Portal) in the Reports section.
The Alert Settings screen is accessed as a sub-menu of the Reports menu. Click on the downward pointing arrow on the Reports item to expand the sub-menu.