Have a look at the JSONRPC docs:
--jsonrpc-port PORT Specify the port portion of the JSONRPC API server
--jsonrpc-interface IP Specify the hostname portion of the JSONRPC API
server, IP should be an interface's IP address, or
all (all interfaces) or local (default: local).
--jsonrpc-cors URL Specify CORS header for JSON-RPC API responses.
--jsonrpc-apis APIS Specify the APIs available through the JSONRPC
interface. APIS is a comma-delimited list of API
name. Possible name are web3, eth, net, personal,
parity, parity_set, traces, rpc, parity_accounts.
--jsonrpc-hosts HOSTS List of allowed Host header values. This option will
validate the Host header sent by the browser, it
is additional security against some attack
vectors. Special options: "all", "none",
So, enabling the required APIs as you did is only one piece of the puzzle you are trying to solve :)
Assuming you are fine with default port and default APIs, let's have a look at the following features you probably want:
--jsonrpc-interface all enables the HTTP JSON-RPC API on all network interfaces, not only the local loop-back. You could also specify
--jsonrpc-interface 22.214.171.124 which your public IP to only serve RPC to the public network interface.
Give this a try, it should work already, assuming you have opened the RPC port against the AWS firewall. Additional fine-tuning can be achieved via
--jsonrpc-hosts all to allow all host header values sent by the browser.
--jsonrpc-cors '*' to allow all CORS requests from any domain.
Note, you probably want to disable sensible APIs such as
personal,parity_accounts,parity_set and also disable other features such as
--no-storage to harden your client.