Assuming you've got the correct tire size for your vehicle, I don't think tire pressure should affect the odometer.
Let's assume your tire is a perfect circle. Since it's a perfect circle, the circumference of the circle is given by the equation 2(pi)r. Now, let's add a level of realism by letting the tire become "egg shaped" because the air pressure is slightly less than that required to maintain a perfect circle shape. Well, whether the tire is a perfect circle or egg shaped, the circumference of the tire is still the same. Well, instead of having some sort of complicated laser measurement system pointed at the surface of the tire in contact with the ground and so being able to calculate the actual distance travelled by the surface of the tire along the road, there's a sensor that detects wheel rotation. So, let's say for example you've got a tire with a circumference of 5.28 feet. So whether the wheel rotates because it's a perfect circle or egg shaped, it still needs to turn 360 degrees for a single rotation. So if the sensor counts that the wheel has rotated 1000 times, then the wheel has travelled 5280 feet or 1 mile. One that 1 mile is travelled, then your odometer value increases by 1 mile.
The tire may stand up taller when inflated more, but the tread doesn't really stretch. The distance around the outside of the tire (which I'm not calling the circumference, since it's not a circle) doesn't really change.
If I wanted to measure it, I'd make a chalk mark on the sidewall and on the ground, and roll the tire 10 revolutions (while mounted to the car - you will need an assistant...) Pump it up, and do the measurement again. Let the air out, and measure again.