Is it necessary that an iOS app has to be built to be used on the parent's device in order to set restrictions and control the activities on the child's device ?
For example, can a web app be built to allow parents to set restrictions on their child's devices ?
A parent iOS app is not necessary to use the ScreenTime API: What's important is that the child device is authorized in FamilyControls for your app. This is necessary to enable the use of the DeviceActivity and ManagedSettings frameworks on the device.
A web app should be possible, provided you have an authorized app or its extensions set the actual restrictions on the child's device.