The relationship described in the first analogy is based on increasing intensity:
\[
\text{dry} \to \text{arid} \to \text{parched}.
\]
This progression moves from a low level of dryness (\text{dry}) to extreme dryness (\text{parched}).
Similarly, in the second analogy:
\[
\text{diet} \to \text{fast} \to \text{\_\_\_}.
\]
Step 1: Analyze the terms. "Diet" implies restricting certain foods. "Fast" intensifies this by avoiding all food for a specific time. The next level of intensity would be "starve," which represents an extreme and prolonged lack of food.
Step 2: Eliminate incorrect options.
- (2) "reject" and (4) "deny" do not align with the context of increasing food restriction.
- (3) "feast" is the opposite of restriction.
Final Answer:
\[
\boxed{\text{starve}}
\]