[Bug] Subagent status line JSON missing parent reference for nested subagent hierarchy

Bug Description
the subagent status line JSON contains no info to match nested subagents to their parents, i.e. it is not possible to display the "<agent-type> (+X)" like the original subagent status line does

Environment Info

• Platform: linux
• Terminal: konsole
• Version: 2.1.238
• Feedback ID: ad8b617d-1cf7-4f94-a63d-f25843606cd3

Output as of 2.1.238:

{
  "session_id": "...",
  "transcript_path": "...",
  "cwd": "...",
  "prompt_id": "...",
  "columns": 169,
  "tasks": [
    {
      "id": "a9694f99227974bfe",
      "type": "local_agent",
      "status": "running",
      "description": "haiku (low) - Infinite block test 1",
      "label": "Backgrounding infinite bash processes",
      "startTime": 1787309508060,
      "model": "claude-haiku-4-5-20251001",
      "contextWindowSize": 200000,
      "tokenCount": 21626,
      "tokenSamples": [
        ...
      ],
      "cwd": "...."
    },
[...]
    {
      "id": "aeee17882615c47fe",
      "type": "local_agent",
      "status": "running",
      "description": "haiku (low) - Infinite block test (nested)",
      "label": "Reinvoking timeout sleep loop",
      "startTime": 1787309763895,
      "model": "claude-haiku-4-5-20251001",
      "contextWindowSize": 200000,
      "tokenCount": 19584,
      "tokenSamples": [
        ...
      ],
      "cwd": "..."
    },
  [...]
  ]
}

Looks like this:

<img width="1466" height="103" alt="Image" src="https://github.com/user-attachments/assets/8436006c-ba60-4bff-9ddc-74f85a69dc76" />