Yours is a high efficency furnace with a draft inducer which blows on a pressure switch to allow the gas valve to open. I don't see how that can be a problem unless the inducer motor is not running. If it is "on" but not running it is a problem.
Open the inducer and clean out any lint or dust that may have accumulated over the years, especially on the inducer fan (or impeller); put it back together and try it again. It also helps to wire brush and vacum the burners
and pilot tube for a better lighting cycle.
It may not have a standing pilot light but rather a spark ignitor or a hot surface ignitor. If it is a hot surface type, be very careful not to crack the ignitor or examine it with a magnifying glass for cracks. They are notorious for cracking after a year of two of heating to a glowing red and cooling down.
Please post back and let us know how it works out.