Can I delete pictures on my old iPhone through my new one using the same iCloud?
If both devices have the same iCloud credentials and iCloud Photo Library is on in both of them, deletions are synced between devices.
The same principle applies to My Photo Stream.

You can even delete the images on iCloud, no need for a second iPhone.

You can download them from iCloud or copy them to a computer from the second device before deleting them.

If you are worried that someone may see them at all, go to the "Recently deleted" album and delete them from there as well.

The iPhone you want images to be remotely deleted on must connect to internet for this changes to occur.