What is the purpose of cryptographically signing request parameters in HTTP APIs?

Only if you require that all your clients perform mutual SSL authentication (i.e. they send a client-side certificate). Without it, SSL only provides a secure channel. With the usual SSL setup (server sends a certificate, client only does verification), the client can authenticate the server, while the server has no idea which client is talking to it. So, you have to provide some way to authenticate clients — either by requiring all clients to send a certificate and go through the hassle of acquiring a certificate, and having the server need to keep a list of allowed certificates etc., or by adding an additional layer where you sign requests. OAuth goes with the latter, having requests signed instead. Of course, as a side benefit, with requests being signed this way you can still be able to authenticate clients outside of an SSL connection (i.e. plain HTTP).

Signing parameters also provide non-repudiation. This is particularly important when dealing with transaction values. This is useful for denials later on such "I did not authorize $100 transfer only a $1 one".