Is there a difference between dp and p on a pinion gear?

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Pitch and diametric pitch are the same thing. Module is based on that but calculates differently. You either are given P/DP or Mod. Those have to match for the gears to mesh.
 
You're tracking. The only thing missing in your 32p example is the # of teeth to determine module. As diameter goes up, the number of teeth go up unless the module changes. Here's a quick wikipedia excerpt. It's pretty simple:

Although gears can be made with any pitch, for convenience and interchangeability standard pitches are frequently used. Pitch is a property associated with linear dimensions and so differs whether the standard values are in the imperial (inch) or metric systems. Using inch measurements, standard diametral pitch values with units of "per inch" are chosen; the diametrical pitch is the number of teeth on a gear of one inch pitch diameter. Common standard values for spur gears are 3, 4, 5, 6, 8, 10, 12, 16, 20, 24, 32, 48, 64, 72, 80, 96, 100, 120, and 200.[38] Certain standard pitches such as 1/10 and 1/20 in inch measurements, which mesh with linear rack, are actually (linear) circular pitch values with units of "inches"[38]

When gear dimensions are in the metric system the pitch specification is generally in terms of module or modulus, which is effectively a length measurement across the pitch diameter. The term module is understood to mean the pitch diameter in millimetres divided by the number of teeth. When the module is based upon inch measurements, it is known as the English module to avoid confusion with the metric module. Module is a direct dimension, unlike diametrical pitch, which is an inverse dimension ("threads per inch"). Thus, if the pitch diameter of a gear is 40 mm and the number of teeth 20, the module is 2, which means that there are 2 mm of pitch diameter for each tooth.[39] The preferred standard module values are 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.8, 1.0, 1.25, 1.5, 2.0, 2.5, 3, 4, 5, 6, 8, 10, 12, 16, 20, 25, 32, 40 and 50.[40]
 
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