Leave the car home and take public transportation. From anywhere along 125th St, you can take the M60 to LaGuardia(A, C, or D train to St. Nicholas Ave; 2 or 3 train to Lenox Ave; Metro North at Park Ave; 4, 5, or 6 train at Lexington Ave; in Queens, the N train at Astoria Blvd.) If taking the LIRR, go to Woodside, then take the Q70 Limited to LaGuardia. The Q70 can also be picked up at the Jackson Height-Roosevelt stop on the E,F, M, R, or 7 trains(the first four trains are underground, the 7 is above ground).
Yes, it is expensive parking a car at LaGuardia and there are no cheap alternatives that do not involve moving one's car on designated days.
Good luck. Hopefully, someone reading this thread will come through for you.