Can your parents see your search history from their phone?

Not from their own phone, no — but they can see it on yours, and that is the route most parents actually use. Search history lives on the device and in the account: Safari history under Settings → Safari → Advanced → Website Data, Chrome under History, Google activity at myactivity.google.com for anything signed in to the account.

What a parent can see without any app:

• On the child's phone, in the browser — Safari history, Chrome history, and the router's DNS log if the home network is managed.
• Through the Google account — My Activity shows searches made while signed in, on any device, including voice searches.
• Through Screen Time — websites visited, but only when the child uses Safari, and only categories/times, not every URL, depending on the setup.
• Through the carrier — DNS logs, if the plan includes family controls.

What they cannot see from their own phone remotely: another device's local browser history. That requires either the account (Google/Apple) or software installed on the device with the user's knowledge.

Where our app fits. Ultimate Phone Spy runs on Android 4.0+, Windows and macOS and reports browsing activity from the monitored device to the dashboard (features). On iPhone there is no equivalent for us — Screen Time and the Apple ID are the honest answers there.