I assume what you want to do is given the file foo.bar you want this run:

awk '{ print $0,"\t","foo" }' foo.bar > foo.bar

But that will not work, as you have foo.bar both as input an output. What will happen is that foo.bar will be 0 bytes long.

So you need a temporary file:

awk '{ print $0,"\t","foo" }' foo.bar > foo.bar.tmp; mv foo.bar.tmp foo.bar

To do that using parallel it might be easier to define a function:

doawk() {
   FILE=$1
   BASE=$2
   awk '{ print $0,"\t","'"$BASE"'" }' "$FILE" > "$FILE".tmp; mv "$FILE".tmp "$FILE"
}
export -f doawk
parallel doawk {} {.} ::: *.txt

This way you avoid double quoting the awk script.

If you want to avoid the function we are looking at:

parallel awk\ \'\{\ print\ \$0,\"\\t\",\"\'{.}\'\"\ \}\'\ {}\ \>\ {}.tmp\;\ mv\ {}.tmp\ {} ::: *.txt

With no info about the output expected, this is what I think you are trying to do:

parallel "awk '{print \$0, "\t", FILENAME}' {} > {}.tmp; mv {}.tmp {}" ::: *input.txt