Can I recover permanently deleted photos from my old iPhone?

Permanent means what it says.

There are many ways of preserving photos and videos. Did you do any of these?

• iCloud photo sync (go to http://icloud.com/photos and log in with your Apple ID to see any synced photos)
• Manual iCloud backup
• Automatic nightly iCloud backup
• iTunes backup
• Transfer your photos and videos to your computer via USB cable
• emailed, texted or messaged the items to friends or family
• Sync to Google Photos
• Sync to some other photo archiving service such as Shutterfly
• Sync to Dropbox
• Sync to box.com
• Sync to Microsoft OneDrive
• Sync to Amazon Photos
• back up to Livedrive
• Backup to any other backup or archiving service
• Transfer them to a Flash drive through the lightning port on your iPhone, using a product such as SanDisk 64 GB iXpand Flash Drive Go - Apple

A final option is if you synced your photos to iCloud, you have a Mac, and you use Time Machine to back up your Mac automatically, you can open Time Machine and restore the photo library to an earlier date. To do this on your Mac open Finder, then click on Pictures in the sidebar and launch Time Machine. Go to a time before the photos were deleted and restore

Photos Library.photoslibrary. (If you had ever changed the name of the photo library restore that file instead).

If the answer is no to all of the above your photos are gone forever.