Step 1: Concept
To identify the specific day, we need information that points to a single day of the week.
Step 2: Analysis of Statement I
He did not visit on Tuesday or Thursday. This leaves Monday, Wednesday, Friday, Saturday, and Sunday as possibilities. Not sufficient.
Step 3: Analysis of Statement II
The father reached home on the day after Wednesday, which is Thursday. Venkatesh visited the park three days before Thursday.
Thursday $- 3$ days = Monday.
Statement II alone gives a unique day. Sufficient.
Step 4: Conclusion
Correct choice is (2).
Final Answer: (2)