What would be the best way for a startup to approach a large company like Airbnb or CouchSurfing about working together?

I myself run a startup and recently started tying up with big co-working companies and online educational providers in India.

So yes, I don't think you should feel anyway less or intimidated by the brands you try to reach out. Based on the patterns I had observed in my conversions -
a. Email to right person is the best way or second best use of LinkedIn inmail
b. Use of introductions

And the most important - you should be able to really articulate well on how they benefit with your partnership.

You need to know you have something that will be of value to them. So, do your homework, read the annual report and look for things in their strategy that you solve. Listen to earnings calls, and correlate you business to needs you hear on those. Pull a strategy together that is a direct use case from their position on how they make money with your help. Then reach out to the CEO, CMO, Strategy leads on LinkedIn and make the connection with the items you KNOW are of interest to them. Expect this to take 9-18 months, and a lot of meetings, cost, and time. You should also have exactly what you want "The Ask" in the deal, and present that earlier than later, because a "NO" answer can be more valuable than a "Maybe".

Best way is to contact the head of sales/marketing department and present your proposal. Second option is through mutual connections if you have. I would say go ahead and try. I suggest you practice your pitch before you call.

One approach...

Create Facebook ad campaign targeting people using both of these companies.

Skip pitching them + go directly to their clients.

You're profits will be higher + marketing way faster/easier.

If you're wildly successful, maybe one of them will buy you out.

Businesses are in the game of making money. Big or small doesn't matter.

If your services are going to either make their lives easier or make them more money. The best way would be to, reach out to a decision maker.

The following 2 worked best for me, when I sold to companies like OLX, Quikr, TVS among others

Cold Emails
Send 5 emails ( 1 every 7 days) per person. Also, when you are reaching out make sure you reach out to at least a 150 of them every month. You would have a better chance of generating a meeting.

References
The best way though would be if some one can connect you directly to the said decision maker.