HP 5.1.1 Printer Software Update - Apple Support
An HP LaserJet P1505 (driver version 1.6.1) was listed in an earlier article mentioned in connection with the above package. However, the text for the download says Not compatible with macOS v12 and newer, so do not expect it to work with macOS Sequoia 15. If the package will not install, modifications have been suggested for HP LaserJet P1005 and other models (community search), but the success rate is somewhat unclear.
If a compatible HP driver for macOS 15 cannot be found, or if another software solution is not possible, you may have to look for a workaround.
Linux drivers are available for the HP LaserJet P1505.
https://developers.hp.com/hp-linux-imaging-and-printing/supported_devices/index
If nothing else works, you could perhaps investigate whether an inexpensive Linux-based Raspberry Pi print server could make a USB-connected HP LaserJet P1505 act in an AirPrint-like way, in order to allow network printing from a modern Mac or iPhone/iPad. This would be experimental. Details about how to set up a Raspberry Pi print server can be found in the posts by Techguyuk in Can’t print with iPadOS17 - Apple Community. Tesserax reported about using a Raspberry Pi 3 Model B+ for an old HP LaserJet printer (in Can I connect my laser printer to my netw… - Apple Community).