Just for the records: for gnuplot 4.6, Christoph's answer is certainly a good way to go.
However, at the time of OP's question, gnuplot 5.0.0 was already available.
gnuplot 5.0.0 (Jan 2015) has the option to define multiple datafile separators, check help datafile separator. With this, the task simplifies to:
Script:
### extract numbers between various separators
reset session
$Data <<EOD
1 (15, 3): dX: -1.619, dY: 3.315, dXSc: 0.981, dYSc: 0.993
2 ( 4,16): dX: -0.540, dY: -0.540, dXSc: 0.992, dYSc: 0.977
3 ( 1.5,10.5): dX: -0.540, dY: -0.540, dXSc: 0.992, dYSc: 0.977
EOD
set datafile separator "(,)"
plot $Data u 2:3 w lp pt 7 lc "red"
### end of script
Result:

Addition:
Actually, you could also set 4 separators, i.e. "(,):" and split your table and write it into a new "clean" table $DataNew. You have to be careful which columns you select, here it would be 1:2:3:6:8:10:12.
Script:
### extract numbers between various separators, get "clean" table
reset session
$Data <<EOD
1 (15, 3): dX: -1.619, dY: 3.315, dXSc: 0.981, dYSc: 0.993
2 ( 4,16): dX: -0.540, dY: -0.540, dXSc: 0.992, dYSc: 0.977
3 ( 1.5,10.5): dX: -0.540, dY: -0.540, dXSc: 0.992, dYSc: 0.977
EOD
set datafile separator "(,):"
set table $DataNew
plot $Data u 1:2:3:6:8:10:12 w table
unset table
set datafile separator # set to default
print $DataNew
# plot whatever you like...
### end of script
Result:
$DataNew
1 15 3 -1.619 3.315 0.981 0.993
2 4 16 -0.54 -0.54 0.992 0.977
3 1.5 10.5 -0.54 -0.54 0.992 0.977