Step 1: Diode Behavior
- Diodes exhibit a non-linear \( I-V \) characteristic.
- Reverse bias (negative voltage) results in near-zero current until breakdown.
- Forward bias (positive voltage) causes current to rapidly increase after the threshold voltage (approximately 0.7 V for silicon diodes).
Step 2: Parallel Diode-Resistor Circuit
In a parallel circuit with a resistor and a diode, both components affect the current-voltage behavior:
1. The resistor provides a linear current increase with voltage.
2. The diode shows a non-linear characteristic: initially small current, then a rapid increase after the forward voltage threshold.
Step 3: Graph Analysis
- Option (A): A steeply rising curve for positive voltage resembles a diode, but may be too idealized for a parallel resistor.
- Option (B): A straight line is incorrect due to the diode's non-linear \( I-V \) characteristic.
- Option (C): A graph with a small hump implies a more complex interaction, less common in a simple parallel diode-resistor circuit.
- Option (D): A sharply increasing graph after a voltage threshold is the most probable diode behavior.
Step 4: Conclusion
The most likely current-voltage characteristic for a parallel diode-resistor circuit starts with small current and sharply increases after a voltage threshold.
Therefore, the correct answer is:
\[\n\boxed{(D)}\n\]