Unfortunately, there’s no built-in button in your iPhone’s phone app for recording calls. Instead, you’ll have to use a third-party app or service or use a clever workaround.
You can delete individual voicemails as you listen to them, and most networks will automatically delete old voicemails after 30 days. However, few networks offer a simple way of deleting multiple voicemail messages when your mailbox is full – unless you can use an option such as Visual Voicemail, which lets you delete messages with a quick tap of your finger.

From the Skype for Business desktop app (if you are running Click-2-Run build C2R 8201.1002 or later), select Set Up Voice Mail. Change your Call Answering Rules You can change what options your callers have when they reach the voicemail service; get disconnected, hear your greeting, record a message, choose to be transferred to the number or user you select. Change your prompt language This is the prompt language that is played to callers. For list of supported languages, see Languages for voicemail greetings and messages from Skype for Business. Configure your Out of Office greeting This is the customized greeting message that is played to callers when your status is Out of Office. This option can be activated "Always," when you have an active "Auto Reply" in Outlook, or whenever you have an out-of-office calendar appointment.  Where you can get your voicemail messages from
If your iPhone has support (via your mobile carrier) for 3-way calling and Visual voice mail, you have an option for the cheapest workaround of all.