This feature has been requested many times. It is one of the most popular requests on the #wishlist!
Follow the link, vote at the top for the request, drop some like hearts around, and add your own reply to the bottom if you want.
This is already possible and can be set in your cam settings. How detailed the settings are will depend on your cam Firmware.
The PanCam V1 (per your topic tag) got the upgraded notification settings some time ago so make sure you have the latest firmware.
In the Event Recording settings, under Smart Detections, toggle on all the CamPlus AI events you want to record and upload as event videos. In Notifications settings, toggle off those recorded AI events that you don’t want notified. You should then be able to record and upload events, but not receive notifications for them.
Last, use the Notifications Settings in the cam to tell the server which push notifications you want from this cam. In your case Pet only. Make sure the Other Motion Events is disabled if you previously chose to record All Motion Events. Otherwise you will be flooded with notifications every time there is an upload.