You need to be a JIRA Administrator in order to create new projects, and you are probably missing that permission. My suggestion is to get in touch with an administrator of your instance and verify if you have that kind of access.
Clarissa is point. I just wanted to add for future reference just have a JIRA admin use [Permission Helper|https://confluence.atlassian.com/display/JIRA/JIRA+Admin+Helper#JIRAAdminHelper-PermissionHelper] It's a great tool for figuring out permission requirements.
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