Does the GPS specs state anywhere how much current it draws?
I've designed a number of devices that were powered from the modem-control lines on the serial port. Here is my experience, FWIW.
Data-Terminal-Ready (DTR) and Request-To-Send (RTS) can be used to power external circuitry, as long as the circuit does not use these signals as modem control.
These lines were designed as signals, not for power, so they are not always reliable.
The software needs to set these lines to the level required by the circuitry, or the circuitry needs to be designed around the levels that the software will set these levels to. The latter is not always feasible. Most of the devices I have built required it's own drivers anyway, so I could set the modem control lines to what I needed.
The voltage is typically around +/-10 volts, but can be as high as +/-15 volts or as low as +/-5 volts. Many laptops put out only +/- 5 volts.
Each line can typically provide 10 milliAmps. If the circuit requires both positive and negative voltage, each rail is limited to around 10 milliAmps. If you only need a positive supply, you can parallel DTR and RTS to get 20 milliAmps.
ALWAYS use series diodes on these two lines to prevent the incorrect polarity from burning things down.
And now, E, for your GPS:
I will assume you need only a positive supply, and that you have no control over what the GPS software does with DTR and RTS.
If your GPS draws more than 20 milliAmps, you may be out of luck.
If your GPS draws less than 20 milliAmps, check the voltage in pin 4 (DTR) and pin 7 (RTS) with your GPS software running. If they are both positive, you are good to go.
If only one of them is positive, you are still good to go if the GPS draws not much more than 10 milliAmps.
Your mileage may vary. I've seen serial port pins supply much more than 10 milliAmps, and I've seen devices draw much less than their specifications state. So the only way to know is to try it. Don't forget the diodes.