I don't think Confluence has a REST API endpoint for that yet, but you can use the XML-RPC/JSON-RPC API for that. There's a method called "addProfilePicture" that you can use: https://developer.atlassian.com/server/confluence/remote-confluence-methods/#user-management
If you're looking to do so in JIRA, it's a bit different and can be done via the REST API. This was the answer I used when I was trying to figure it out: https://community.atlassian.com/t5/Answers-Developer-Questions/Avatar-upload-with-Jira-5-REST-API/qaq-p/475376
The following REST calls may help you. These calls are private. I'll try them tomorrow, too. Let's see what happens. Or have you already solved it?
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