Step 1: Recall the effect of substituents on acidity.
Acidity of phenols depends on the stability of the phenoxide ion formed after loss of proton.
Electron-donating groups decrease acidity because they destabilize the phenoxide ion.
Step 2: Analyze compound I.
Compound I is phenol. It has no electron-donating substituent, so it shows the highest acidity among the three compounds.
Step 3: Analyze compound II.
Compound II contains a methyl group \((\mathrm{-CH_3})\), which shows \(+I\) effect and donates electrons.
This decreases the stability of phenoxide ion and reduces acidity.
Thus,
\[
\text{Phenol} \gt p\text{-cresol}
\]
Step 4: Analyze compound III.
Compound III contains a methoxy group \((\mathrm{-OCH_3})\), which strongly donates electrons by resonance effect \((+M)\).
This greatly destabilizes the phenoxide ion, making it the least acidic compound.
Therefore,
\[
p\text{-cresol} \gt p\text{-methoxy phenol}
\]
Step 5: Write the final order.
Hence, the acidity order is
\[
I\gt II\gt III
\]
Step 6: Final conclusion.
Therefore, the correct order of acidity is
\[
\boxed{I\gt II\gt III}
\]