Well it would seem that you can do this, via the "JSON-RPC API," which I have no experience with. So at a very beginner level, I can tell you what I have figured out:
1. You need to start Parity with the API features enabled:
parity --jsonrpc-apis web3,parity,parity_accounts,personal
Next, structure a command for the "personal_sendTransaction" command, which includes the to/from addresses, and an unencrypted password for the "from" account. You can read about the APIs here: https://wiki.parity.io/JSONRPC-personal-module.html#personal_sendtransaction
Just for a safe little test to see if you can use the API, just try to list your accounts, or create a dummy account:
curl --data '{"method":"personal_listAccounts","params":[],"id":1,"jsonrpc":"2.0"}' -H "Content-Type: application/json" -X POST localhost:8545
AND
curl --data '{"method":"personal_newAccount","params":["hunter2"],"id":1,"jsonrpc":"2.0"}' -H "Content-Type: application/json" -X POST localhost:8545
I hope this was informative for someone. Thanks!