Yes, this is possible though probably not as straightforward as you want it. I don't believe there's a "bulk copy" type feature for copying all parameters and values between projects, but you can modify the schematic template to ensure that the proper parameters (and values) are populated when you create a new project.
First, find where the schematic template file you're using (or want to use) is located and open it for editing. For example, I'm using a custom schematic template for an 11x17 piece of paper called 11x17.SchDot
Open the document options by selecting
Design > Document Options and clicking the "Parameters" tab as shown below.
I added a value called "TestValue" by clicking the "Add..." button and entering my desired parameters.
Then, I can open a schematic (or create a new schematic) and if this is the default schematic template for new schematics, it should already have the "TestValue" parameter in the document options. If it's not the default, or if you want to apply the template changes to an existing schematic/project, select
Design > Templates > General Templates > Choose Another File... and load the template that you applied parameter changes to. You should see the following dialog pop up:
You can apply the template changes to all schematic documents in the current project, and likewise, you can choose to either only add new parameters that are in the template but not in the current document, or you can replace each matching parameter with the specified value.
If this is something you do for every new project, I suggest you copy the desired schematic template file from the default location to a common/custom location and give it a name that will differentiate it from the defaults (e.g., COMPANY_NAME_A4.SchDot) Then, in your Altium Designer preferences, ensure that the defaults are configured to use this template in new projects.