No app can read another iPhone's SMS. Apple does not expose the Messages database to third-party software, and there is no non-jailbreak product that changes that — the ones claiming it are selling an iCloud backup view or nothing at all.
What a parent can actually see on iPhone. Screen Time's Communication Limits control who your child can message and call during downtime — the allowed-contact list — but not the content. If the family uses one Apple ID for Messages (common with younger children), messages appear on every device signed into that account, including yours; that is Apple's own sync, not monitoring. Family Sharing with "Ask to Buy" and location sharing covers purchases and whereabouts.
If you hold the Apple ID and want the backup view, services exist that read iCloud backups and surface message history, photos and some app data on a delay. Two honest caveats: backups only contain what has synced, and two-factor authentication alerts the account holder.
Where our app fits, honestly: not on iPhone. Ultimate Phone Spy supports Android 4.0+, Windows and macOS (supported devices). On Android it reads SMS and MMS directly, with the child's knowledge, alongside call logs, WhatsApp and Snapchat notifications, and location. If the child's phone is an iPhone, we are the wrong product and I would rather say so than take the sale.