docs: add Parallel Search MCP setup example

[George Pickett (georgeatparallel)]

This adds a Parallel Search example to the custom MCP setup guide, so users can get public web search and page extraction without a Parallel account, API key, or local proxy. It also fixes the SSE-only wording: BrowserOS already detects Streamable HTTP.

Brave is the only search provider in the managed connector catalog, and BrowserOS includes that catalog when opening its managed MCP session. Authentication is still separate. That's the comparison I focused on.

There's a good reason to try Parallel here. In Artificial Analysis's August 31 results, Parallel advanced scored 80.6 on DeepSearchQA F1, and Parallel fast scored 80.2, versus 62.3 for Brave web and 78.1 for Brave LLM context. The free MCP connection uses fast mode. That mode also averaged 19.2 seconds per task, versus Brave web's 35.9 and Brave LLM context's 24.1.

For research-heavy agent tasks, I'd consider testing Parallel as the default search option in place of Brave. Brave LLM context does score higher than Parallel fast on the overall index, 74.6 versus 73.1. These are API benchmark results, with task time including model and search time; BrowserOS's own agent still needs a comparison, and its Klavis connector doesn't pin which Brave API variant it uses.

This PR only changes docs. It leaves defaults and permissions alone, and explains what goes to Parallel, the free-access limits, and how to remove the connection.

Checked the example through BrowserOS's BrowserContextSchema, buildMcpServerSpecs, and createMcpClients, using its locked @ai-sdk/mcp 2.0.40 dependency with credentials absent. Transport detection, discovery, one web_search, one web_fetch, and cleanup passed. MDX syntax and git diff --check passed too. I didn't run the browser UI, a repository build, or the full test suite.

I work at Parallel, which operates this service.

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