The example in RPC help is:
curl --user myusername --data-binary '{"jsonrpc": "1.0", "id":"curltest", "method": "sendmany", "params": ["", "{\"1D1ZrZNe3JUo7ZycKEYQQiQAWd9y54F4XX\":0.01,\"1353tsE8YMTA4EuV7dgUXGjNFf9KpVvKHz\":0.02}", 6, "testing"] }' -H 'content-type: text/plain;' http://127.0.0.1:8332/
Which is wrong: the second argument "{\"1D1ZrZNe3JUo7ZycKEYQQiQAWd9y54F4XX\":0.01,\"1353tsE8YMTA4EuV7dgUXGjNFf9KpVvKHz\":0.02}" needs to be a JSON object, not a JSON object serialized as string, I think this would be correct:
curl --user myusername --data-binary '{"jsonrpc": "1.0", "id":"curltest", "method": "sendmany", "params": ["", {"1D1ZrZNe3JUo7ZycKEYQQiQAWd9y54F4XX":0.01,"1353tsE8YMTA4EuV7dgUXGjNFf9KpVvKHz":0.02}, 6, "testing"] }' -H 'content-type: text/plain;' http://127.0.0.1:8332/