Is airfoil thickness proportional to drag coefficient?
There are several reasons for using thick$^*$ airfoils.


Slow planes need high lift coefficients in order to fly slow. The increased drag coefficient is the prize to pay for a higher $C_{L,max}$. Take a look at this diagram:




Source: H. Schlichting, E. Truckenbrodt, Aerodynamik des Flugzeuges. Colors added.

At low Reynolds-Numbers the maximum lift coefficient is achieved by the thickest airfoil (18%). At slightly larger Re-Numbers the 12% thickness airfoil gains the upper hand. However, the thinnest airfoil (9%) has on average about 20% less maximum lift capability than the thicker ones.

The following diagram shows polars of symmetric airfoils with different thicknesses.



Source: B. Rögner, Flugwissen. I couldn't find the original source Rögner took the diagram from, if anybody knows it feel free to comment. 

The quality of image is bad, but you can recognize both the higher $C_{L,max}$ and drag coefficient of thick airfoils.

The drag coefficient depends on many factors and, as shown in the formula given in Peter's answer, is not proportional to the thickness. The drag coefficient's thickness dependency however, can be assumed to be proportional to the thickness: the $\delta^4$ term is one order of magnitude smaller than the term multiplied with $\delta$.

$$\frac{c_{D,\text{normal airfoil}}}{c_f}=2+4\cdot\delta+120\cdot\delta^4$$

$$\frac{c_{D,\text{laminar airfoil}}}{c_f}=2+2.4\cdot\delta+140\cdot\delta^4$$     

In Hoerner's Fluid Dynamic Drag, Chapter 6.A.2, you can find more formulas to approximate the drag coefficient and take other parameters into account.

A good resource for understanding the influence of airfoil parameters on their aerodynamic properties is NACA TM 824.


Stall behavior of thicker airfoils is usually more forgiving.  
Another reason for using thick airfoils is that it decreases the wing's weight, as thick structures can carry better the bending loads in a wing. If a wing tank is used, thicker airfoils allow for a higher tank volume.




$^*$: I interpret thick as >12%, e.g. 15% as found in the wings of many GA airplanes.