You don't need to install pse6 on the new computer. You only need the new and old serial numbers at the time of sign-in and activation.
>> if I complete the upgrade purchase, will that give me the ability to download the full application of PS Elements 2020 onto my new Mac? Or will it be looking for the old software
The Elements 2020 installer is always the full version of the software. Upgrading is all about the serial number. The serial number you enter tells the installer if you've got a full version or an upgrade version.
As 99jon says, you don't need your earlier version installed at all. You can install an upgrade even on a brand new machine untouched by Adobe software.
When you enter an upgrade serial number, the installer first looks on the hard drive to see if a previous qualifying version is already installed (e.g. you cannot upgrade an Edu version, only a retail version). If not, a dialogue box will appear prompting you to enter a serial number from a previous qualifying version (in your case PSE version 6).
>> I'd like the upgrade price rather than having to pay $20 more to start all over again.
There are tradeoffs with that.
For an upgrade version, you will always need your PSE6 serial number in order to reinstall your PSE 2020 upgrade version. Many people lose track of the qualifying version serial number and that leaves them unable to install 2020. Having a full version means you only need one serial number to reinstall.
An upgrade version also means that:
If you buy PSE 2020 full version then you have two licenses and you can sell PSE6 and PSE 2020 separately.
These may not apply to you now or ever. Just things to be aware of when you make your purchase. If you prefer to just keep the costs down then buy the upgrade.
Thanks to both reply posts, you have answered my question. It never occurred to me to try to sell PS6 with PS2020 upgrade but great to know that's an option.