You are correct, this is not currently an available function when it comes to the Search field. Your change request was logged. ID: CR-248562.
As a side note, you can easily work around this missing functionality by filtering by the desired Parent Project. You can do this in two ways...
1 - Add the Parent Project field to your view (Click on Gear Icon --> Manage --> Select desired field(s) to view. Then you can sort by the Parent Project Column.
2 - Add the Parent Project field to your filter (Click on the Filter Icon --> Manage --> Select desired field(s) to filter on. Then you can filter by the Parent Project field in the filter. You can even save as your default view by checking the 'Save as default' checkbox near the Save button.