Frances - although Joshua's guess makes sense, it is not the case. There are two different packages containing the coordinate system data. The screenshot below is from the My Esri download page for Pro 2.6 but it has been that way for a while.
The one you are dealing with is ArcGIS Coordinate Systems Data for ArcGIS Pro Per User Install, and by design installs into the user's Documents folder with the logic that only that user will be running ArcGIS Pro since it is licensed to the user.
If you download the other one (ArcGIS Coordinate Systems Data) from the ArcGIS Pro section of My Esri when you unpack it, it will unpack into the ArcGIS Desktop 10.8.1 folder. This one installs by default into
C:\Program Files (x86)\ArcGIS\CoordinateSystemsData\
If you are also installing ArcGIS 10.x then both ArcMap and Pro will use the data from that location.
If you don't like it there, you can tell the ArcGIS Pro specific installer where to put it. Here is the line from my installation script to put it in C:\ProgramData\Esri.
msiexec /i "%~dp0ProCoordinateSystemsData\ProCoordinateSystemsData.msi" INSTALLDIR="C:\ProgramData\Esri" /qn
A man's feet should be planted in his country, but his eyes should survey the world. - G. Santayana