If the Earth is curved, why don't planes need to adjust attitude to stay parallel to the ground? If I'm flying parallel to the ground and I never adjust my attitude, shouldn't my altitude above the ground start rising if the Earth is curving away from me? In practice, why doesn't this happen. 
Edit: Follow up question to the answers already given. 
Most have suggested that the plane's course is indeed corrected. If we ignore the fact that the atmosphere is not uniform, would it be possible to have the lift cancel out the gravitational force just enough that the conditions for circular motion are met i.e.
$$F_g-F_l=\frac{mv^2}{r}$$?
 A: I am a student pilot. I've got 100 hours and 300 landings.
Throw a paper airplane. Does it fly in a straight line? Of course not.
Neither does a big plane.
It only follows the course, horizontally and vertically, that the pilot controls it to fly.
The way it turns from one heading to another is by temporarily banking to one side, exactly like a bicycle.
The way it changes altitude above the ground is by using more or less power.
The way it changes speed is by using more or less elevator trim.
All of these things are monitored by the pilot and adjusted as needed.
If they are not adjusted, the plane flies at a constant speed, on a constant heading, at a constant rate of climb or descent above the earth.
Take a flying lesson. It doesn't cost much and is a great experience.
A: I'm no pilot, but I believe that airplanes typically (whether on autopilot or not) maintain a constant altitude when cruising, i.e. height above the ground, using an altimeter. So if the plane (via computer or human operator) detects the altitude is increasing, the plane uses its control surfaces to alter its ascent rate and go back to the desired altitude.
This results in following the curve of the Earth without the need for worrying about whether or not the plane is traveling in a "straight line".
A little geometry leads me to calculate that a 747 at cruising speeds that was intentionally flying in a straight line would ascend about $1.7 ~\rm ft$ in $1 ~\rm sec$, and the plane's nose would be $0.002°$ above the horizon. Now, that ascent rate would increase nonlinearly if the plane continues in that direction, but presumably the pilot or autopilot would be continuously adjusting the plane's attitude, so the ascent rate would never get much higher than that.
A: 
If I'm flying parallel to the ground and I never adjust my attitude, shouldn't my altitude above the ground start rising if the Earth is curving away from me?

This question, and most of the responses and comments to it, are based on a misunderstanding of what "straight and level flight" means. It does not mean flying in a Euclidean straight line at a constant speed. The concept of flying a Euclidean straight line doesn't make any sense when it comes to an aircraft flying through the atmosphere of a more or less spherical planet. "Straight and level" instead means flying with zero bank, zero roll, and a constant speed, constant altitude, and constant pitch angle.
The pitch angle of an aircraft is measured with respect to local vertical / local horizontal (LVLH). This means that from the perspective of an inertial observer, an aircraft in straight and level flight is constantly pitching down. The only people to whom this is a concern are the designers of an aircraft that uses an Inertial Navigation System. Even then, the first thing the designers of such a system will do is to transform the inertial attitude and attitude rate into quantities more amenable to a human pilot or to an automated control system. Those more amenable quantities include attitude and attitude rate with respect to LVLH.
An LVLH frame (e.g., north-east-down or east-north-up) is a rotating frame. There are applications where an inertial frame is less useful than is a rotating frame. Flying an aircraft is one of them.
A: The answers I've seen above accept your premise: the plane would indeed fly off-Earth if it was "dumb" (i.e. altitude not controlled by human or software). This is not the case.
This is my attempt to explain as simply as you are reasoning: 
Assume a "dumb" plane flights straight, parallel to the current position. Nothing turbines pumping and there is no way to change altitude (all flaps fixed). This airplane would not escape the Earth, why?
First if the turbines work worse as air rarifies at high altitudes, and at some point lose "traction" and fall. Secondly, the levitating force on the wings depends on the air density and the plane relative speed to the air, so even if the "traction" effect is low, the air support will be lower and at some point will not be sustained. Thirdly, as it escapes horizontally the angle of vertical becomes that of the flight direction, or as was mentioned, it will behave as if you suddenly start flying up, which will eliminate air support on the wings and fall.
So in summary, even a "dumb" plane will not escape horizontally because is much like a boat in the ocean: its stays on floating altitude rather than falling out of the horizon. It depends on the medium to perform, it is propelled by the medium (even turbine planes) and is supported by the medium. 
BTW, that is also why we cannot use turbines to fly rockets up to vacuum space.
A: In practice, this is exactly what would happen if you were to fly in a straight line.
This is why planes don't fly in a straight line: they adjust their path to stay at the same altitude.
