You can't do this with parameters, but you can do it with quick filters with "Only Relevant Values" option selected.
Just add the Country, State and District dimensions to the Filter Shelf. Right-click each > Show Quick Filter. You right-click the filter box and change this to a single-select pull-down. And also enable the Only Relevant Values option.
If you're filtering this works reasonable well, with the main issue being that once you select the District level, the middle filter will change from the state name to (All) since you are technically showing all values.
Another issue is that clearing the filters isn't very clear. You might need to use a dummy worksheet called "Reset Selection" and a dashboard action to accomplish this.
If you need to use these values in calculations, you may need to perform a check to see if only one value is selected. Something like IF MIN(Country) != MAX(Country) THEN "Select a country." ...
As with many things in Tableau, the best approach depends on your data, goals, and dashboard layout. If the above doesn't solve your issue, I'd recommend posting a sample packaged workbook (twbx)---if you don't want to share your data, you can use the Super Store as an example.