TIMTOWTDI!

while (<DATA>)
{
    m/^(\w+)\s+(\w+)/;
    push (@array, $1) unless $2 eq "Veterinarian";
}

print("@array\n");
__DATA__
David    Veterinarian
Jackie  Orthopedist
Karen Veterinarian
[download]

or even

while (<DATA>)
{
    @_ = split /\s+/;
    push (@array, $_[0]) unless $_[1] eq "Veterinarian";
}

print("@array\n");
__DATA__
David    Veterinarian
Jackie  Orthopedist
Karen Veterinarian
[download]