Caveat: this may not meet your needs if you have to use the exception mechanism.
If I understand you correctly, you don't actually need the exception to be checked (you've accepted the answer suggesting an unchecked exception) so would a simple listener pattern be more appropriate?
The listener could live in the parent thread, and when you've caught the checked exception in the child thread, you could simply notify the listener.
This means that you have a way of exposing that this will happen (through public methods), and will be able to pass more information than an exception will allow. But it does mean there will be a coupling (albeit a loose one) between the parent and the child thread. It would depend in your specific situation whether this would have a benefit over wrapping the checked exception with an unchecked one.
Here's a simple example (some code borrowed from another answer):
public class ThingRunnable implements Runnable {
private SomeListenerType listener;
// assign listener somewhere
public void run() {
try {
while(iHaveMorePackets()) {
doStuffWithPacket();
}
} catch(Exception e) {
listener.notifyThatDarnedExceptionHappened(...);
}
}
}
The coupling comes from an object in the parent thread having to be of type SomeListenerType
.