Unless I am missing something this should do it:

print "First:\n";
print "$_\n" for grep { /[^\s]/ } param("Data_One");

print "Second:\n";
print "$_\n" for grep { /[^\s]/ } param("Second_Part");
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Note: In the grep I only test if the value contains something different than spaces. That logic is flipped to your code, but more natural for grep. It should do the same.