Fix routing of background thread output when no parent is set explicitly #3653

Job	Run time
Test Lint	57s
Test Minimum Versions	2m 46s
test_docs	49s
test_without_debugpy (ubuntu-latest, 3.10)	2m 16s
Test Prereleases	2m 39s
Make SDist	32s
link_check	3m 44s
check_release	1m 40s
build (ubuntu-latest, 3.14t)	1m 58s
build (ubuntu-latest, 3.12)	4m 42s
build (windows-latest, 3.10)	7m 30s
build (ubuntu-latest, pypy-3.11)	3m 43s
build (windows-latest, 3.12)	7m 55s
build (ubuntu-latest, 3.10)	4m 56s
build (windows-latest, 3.14t)	2m 46s
build (macos-latest, 3.10)	5m 21s
build (windows-latest, pypy-3.11)	7m 27s
build (windows-latest, 3.11)	6m 10s
build (ubuntu-latest, 3.11)	4m 41s
build (macos-latest, 3.14t)	4m 1s
build (ubuntu-latest, 3.14)	2m 45s
build (macos-latest, 3.13)	4m 59s
build (windows-latest, 3.14)	5m 18s
build (macos-latest, 3.11)	5m 21s
build (macos-latest, pypy-3.11)	4m 35s
build (macos-latest, 3.14)	2m 35s
build (windows-latest, 3.13)	7m 36s
build (macos-latest, 3.12)	4m 51s
build (ubuntu-latest, 3.13)	4m 53s
Install from SDist and Test	2m 30s
coverage	0s
	2h 1m 56s

Provide feedback