Short answer: most off‑the‑shelf parental apps show contact names, timestamps, and app usage, but not full Instagram DM content unless you have the kid’s device or account credentials (or a rooted/jailbroken phone — avoid that). Least intrusive, effective steps: 1) talk and set rules; 2) use Instagram’s parental supervision tools and privacy settings; 3) follow them or ask to share screenshots; 4) use Screen Time (iOS) / Family Link (Android) to limit app hours; 5) review followers/following and report concerns.
If you want a simple monitoring app that focuses on metadata (not invasive hacks), consider lightweight options like Spynger.