add with_raw_response to responses.parse for OpenAI and AsyncOpenAI

[Barrett Pyke (barrettpyke)]

Context

If you use client.responses.parse through the AI Proxy and include the x-bt-use-cache header it's supposed to add a metric called cached that indicates the request was either a HIT or MISS. client.responses.create does this because it includes the raw response headers but client.responses.parse does not.

Description

Copied the pattern already present for create and added with_raw_response.

Steps to Reproduce:

1. Run the following code:

init_logger(project="barrett-onboarding")

client = wrap_openai(
    AsyncOpenAI(
        base_url="https://staging-api.braintrust.dev/v1/proxy",
        api_key=os.getenv("BRAINTRUST_API_KEY"),
        default_headers={"x-bt-use-cache": "always"},
    )
)


async def main():
    resp = await client.responses.parse(
        model="gpt-5-mini-2025-08-07",
        instructions="you're a storyteller",
        input="tell me a story",
        prompt_cache_key="user",
    )


if __name__ == "__main__":
    asyncio.run(main())

2. Navigate to the named project.
3. Open the Logs.
4. Open the trace that was sent.

Testing

Tested the change locally with the above repro script.