Images uploaded to the Stack Exchange Imgur account are never deleted, and a log is kept of profile image changes made on a user account, which moderators can access. The log records profile image URLs and for Imgur images that means moderators can access the old images still.
So no, old profile pictures are not deleted, provided you used imgur.
Your previous profile picture on Stack Overflow (where I am a moderator) was a gravatar image, which you can change at any time. The logged URL for the image now produces a standard geometric image. The last time you uploaded a profile picture to imgur was in 2016, at least according to the logs on your Stack Overflow account. I assume it was this image you wanted to remove, I think it is still shown as an option next time the gravatar option until you replace it with a different upload.
While it is indeed a personal picture, I’ll not produce the URL here as I am not sure if you’d want the URL to be public at this time or if Imgur honours deletion requests for the SE Imgur site from outsiders.
Since you can set different profile images per site in the Stack Exchange network I can’t see if you used imgur on other sites where I am not a moderator. Only employees can see that level of detail.