Algorithm for determining balanced parentheses
To understand why the answer is option 'C', we first need to understand the algorithm for determining whether a sequence of parentheses is balanced. The algorithm is as follows:

1. Initialize an empty stack.
2. Scan the expression from left to right.
3. If the current character is a starting bracket (‘(‘ or ‘{‘ or ‘[‘), push it to the stack.
4. If the current character is a closing bracket (‘)’ or ‘}’ or ‘]’), then pop from the stack and check if the popped bracket is a matching starting bracket or not.
5. If the bracket is not a matching starting bracket, then the expression is not balanced.
6. After complete traversal, if there is some starting bracket left in the stack, then the expression is not balanced.

Maximum number of parentheses on the stack
Now, let's analyze the sequence of parentheses given in the question: (()(()())(()))

1. The first character is ‘(‘, so we push it to the stack. Stack: (.
2. The second character is ‘(‘, so we push it to the stack. Stack: ((.
3. The third character is ‘)’, so we pop the top element from the stack. Stack: (.
4. The fourth character is ‘(‘, so we push it to the stack. Stack: (().
5. The fifth character is ‘(‘, so we push it to the stack. Stack: (().
6. The sixth character is ‘)’, so we pop the top element from the stack. Stack: (.
7. The seventh character is ‘(‘, so we push it to the stack. Stack: ((.
8. The eighth character is ‘)’, so we pop the top element from the stack. Stack: (.
9. The ninth character is ‘)’, so we pop the top element from the stack. Stack: empty.
10. The tenth character is ‘(‘, so we push it to the stack. Stack: (.
11. The eleventh character is ‘)’, so we pop the top element from the stack. Stack: empty.
12. The twelfth character is ‘)’, so we pop the top element from the stack. Stack: empty.

So, the maximum number of parentheses that appear on the stack AT ANY ONE TIME is 3, which occurs when we are analyzing the subsequence ‘(()())’.

Therefore, the correct answer is option 'C', which is 3.