In QGIS, a tool called "Random Points Inside Polygons" can be used to generate the random points.
Use a point count equal to the number of simulations you plan to run. Optionally, use a minimum distance between points that keeps the points spaced out slightly, although too large of a spacing may prevent QGIS from placing all your points inside the domain. A value close to the resolution of the precipitation data may be appropriate, although with a small transposition domain and coarse precipitation data, not all the points may be able to fit and QGIS will stop generating them.
After creating the points, the Field Calculator can be used to add the X and Y coordinates to the points.
Create new fields for X and Y of type "Decimal number (real)". To access the point's x coordinate, use the expression $x and use $y for the y coordinate.
After adding both fields, save the edits and exit edit mode.
Finally, export the points to a csv file by right clicking on the generated points and selecting Export > Save Features As...
In the options, choose the Comma Separated Value (CSV) format. Expand the "Select fields..." option and ensure only the X and Y fields are toggled on.
Once you have the CSV file, you can open it in spreadsheet software and paste the column into the table for the Parameter Value Samples.