Yes, parallel HTTP connections requires separate three-way handshakes.
Multiple HTTP requests can be handled in one TCP connection. This can happen sequentially, but since HTTP/1.1, the concept of pipelining allows sending multiple HTTP requests in one TCP connection without waiting for the responses. This results in a seemingly parallel handling of HTTP requests, but note HTTP/1.1 requires the server to send the responses sequentially and in the same order as it received the corresponding requests.