Your if (convert == s1.length()) is very strange. That expression is comparing an uninitialized char variable to the length of the s1 string (which will be of size_t type). What you should be doing is comparing the i variable to the length (and that i would be better defined as size_t type).
Also the static_cast <int> (convert); line does nothing. You can't change the type of a variable like that; instead, you should use the cast on the operand of the cout << in the following line. And note that, when casting a character digit to its integer value, you need to subtract the value of '0' from it.
Here's a fixed version that addresses the above issues:
#include <iostream>
#include <string> // You 'forgot' this required header
size_t i = 0; // The "size_t" type is better suited for container sizes and lengths
void parseToInteger(std::string s1)
{
    char convert;
//  std::string another; // Never used!
    if (i == s1.length()) { // Here, we've reached the end of the string
        std::cout << std::endl;                 // prints nothing
    }
    else {
        convert = s1.at(i++);
        std::cout << cout << static_cast <int> (convert - '0'); // Apply the cast here
        parseToInteger(s1);
    }
}
int main()
{
    std::string s0;
    std::cout << "Enter a string to convert it into its integer value: ";
    std::getline(std::cin, s0);
    parseToInteger(s0);
    return 0;
}
But note, there are other concerns in your code, as mentioned in the comments. First, it would be better to pass the s1 argument by reference; second: Are global variables bad?
Here's a version of your function that addresses these last two concerns:
void parseToInteger(std::string& s1, size_t i = 0)
{
    if (i == s1.length()) { // Here, we've reached the end of the string
        std::cout << std::endl;        // 'flushes' output stream
    }
    else {
        char convert = s1.at(i++);
        std::cout << static_cast <int> (convert - '0'); // Convert to int
        parseToInteger(s1, i);
    }
}
The default value (of zero) for the new i argument means that you don't have to change the call in the main function.