Objective:

To record a payment on a bill for a Disbursement directly from the Client/Trust Fund to a Third Party
The payment needs to be on a bill but does not go to the Firm Office/Operating Account

Environment:

To record a payment to a 3rd Party from Client/Trust Funds that does not need to be recorded on a Bill you can do the following: Disbursing Trust and Client Funds
To Create a Disbursements only Bill see here: How Can I Create an Expense Only Bill?

There is not currently an option in Clio to record a payment for a Disbursement on a Bill from the Client/Trust Funds to a 3rd Party.

A workaround for this is to create a second Operating account to record these Transactions instead, usually named Purchase Account

Create a second account:

Go to the Accounts tab and click New.
Account Type: Choose the Account Type Operating.
Account Name: Name your account Purchase Account so that you can identify it in a list.
Optional account information: The bank account information is not required
Opening Balance: You should leave the opening balance at $0.00
Default Account: Leave this unchecked as we want our Office or Client/Trust Account to remain the Default
Select Create New Bank Account

Next, you will need to create a Disbursements only Bill for Expenses that need to be paid in this manner

Then, Record a Payment on the Bill:

When the Bill has been approved Click Record Payment
Select a Source: You will select the Client/Trust Funds
Select a Destination: Here you will select the Purchase Account we created earlier
Enter the Payment Amount
Click Record Payment