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Tracing the code starting from main(), I didn't get very far until I noticed concerns.

Looking at your Expression constructor...

Expression::Expression(std::string expression, bool sub) //NOTE: being a sub means that it was surrounded
//by brackets '(' ')'
{
 hasBrackets = false;
 expressionString = expression;
 containsBrackets();
 // //std::cout << "Successfully complete 'containsBrackets' function" << std::endl;
 if (hasBrackets == true)
 {
 getParentheses();
 // //std::cout << "Successfully complete 'getParentheses()' function" << std::endl;
 getSubExpressions();
 // //std::cout << "Successfully complete 'getSubExpressions()' function" << std::endl;
 }
 evaluate();
 // //std::cout << "Successfully complete 'evaluate()' function" << std::endl;
}

• Use initialization lists.
• Your comment says, "being a sub means that it was surrounded by brackets '(' ')'". Why should that matter though? Indeed, you never even use the sub parameter.
• hasBrackets = false; containsBrackets(); if (hasBrackets == true) { ... } is combersome. Why not write if (hasBrackets = containsBrackets()) { ... }? But even that is unsatisfactory...

Looking a containsBrackets() and getParentheses()...

void Expression::containsBrackets()
{
 for(unsigned int index = 0; index < expressionString.size(); index++)
 {
 if (expressionString[index]=='(' ||expressionString[index]==')' )
 {
 hasBrackets = true;
 }
 }
}
void Expression::getParentheses() //Finds the parentheses and their positions in the expression
//so that their contents can be converted to sub expressions.
{
 for (unsigned int index = 0; index < expressionString.size(); index++)
 {
 if (expressionString[index] == '(' || expressionString[index] == ')')
 {
 Parenthesis temporary(index, expressionString[index]); //Stores the position and type of the parenthesis
 vector_parentheses.push_back(temporary);
 }
 }
 /* ... */
}

• The essentially do the same work. hasBrackets is essentially the same as !vector_parenthesis.empty().
• Do you want to call them "brackets" or "parentheses"? Make up your mind!