It seems to stop the parent too. I don't know if this is the proper workaround, but this works:

use Time::HiRes qw( sleep );

my $pid = fork();
die $! if !defined($pid);

if ($pid == 0) {
    for (1..8) {
        sleep(.250) if $_ != 1;
        print(time(), "\n");
    }
} else {
    $SIG{TTOU}='IGNORE';
    sleep 1;
    kill TTOU => $pid
        or die $!;
    sleep 3;
    kill CONT => $pid
        or die $!;
    waitpid($pid, 0);
}
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And it also works if the child is the one to send the signal.

use Time::HiRes qw( sleep );

my $pid = fork();
die $! if !defined($pid);

if ($pid == 0) {
    for (1..2) {
        kill TTOU => 0
            or die $!
            if $_ != 1;
        for (1..4) {
            sleep(.250) if $_ != 1;
            print(time(), "\n");
        }
    }
} else {
    $SIG{TTOU}='IGNORE';
    sleep 4;
    kill CONT => $pid
        or die $!;
    waitpid($pid, 0);
}
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