Help with OS update

is the MBP an early or late 2013? if it's an early 2013 it does not qualify.

if it's a late 2013 please see these two articles:

How to upgrade to Big Sur - User Tips

How to upgrade to macOS Big Sur - Apple Support

arron293 wrote:

I am operating a 2013 macbook pro on OS 10.10.5 but want to update to the latest but the app store will not let me. Please advise

If you are currently running 10.10.5 you are woefully behind— and will have to stair step your way up.

If you can down load and install at least El Capitan or High Sierra first—then you can move up to the current macOS Big Sur if you have a late 2013 MBP, otherwise macOS Catalina is the end of the line for your hardware.

ref: >About this Mac

The older macOS installer backups on this server:

https://support.apple.com/en-gb/HT211683

verify you are using the Safari browser to download/initiate the macOS links.

note: The macOS 10.11 El Capitan. Includes an improved Mac App Store app and is recommended as a step up to install later versions.