Photoshop CS5 Extended

Keep your old laptop.  We cannot help you with Student/Teacher or volume licensed editions.  Sorry.  You need to use the original media you obtained at purchase.  Creative Suite is no longer legally sold by any vendor, retailer or country now.

OPTIONS:
Photoshop or Premier Elements 2020 (approx $100, no subscription needed)
https://www.adobe.com/products/photoshop-elements.html

Creative Cloud Photography Plan (approx $10/month for 1 year subscription)
https://www.adobe.com/creativecloud/photography.html

Creative Cloud All Apps Plan (31 + apps & services)
https://www.adobe.com/creativecloud/plans.html

Are the installation files not on your old PC? If they are, then you should be able to copy them over and reinstall.

If you bought it as downloadable media from Adobe, it's possible you can still find a download link for it from your Order history page (https://account.adobe.com/orders) by clicking on the link on the right of the page.

If it isn't there, then you're out of luck. My understanding is that Adobe is now only providing downloads for items purchased within the last 3 years and while it seems to be allowing a grace period, there's no telling when that will end.

Be very wary of anyone contacting you via private messages or trying to connect with you via Skype and claiming to be a representative of Adobe with a special offer to rebuy the media. They are scammers. Adobe is not selling this software any more and there is no legitimate method of purchasing it.