Great work! I especially like the ability to colorize text parts!
You really have got a lot of possibilities with your class.
Nevertheless, I have to comment some important things:
Take care of const-correctness.The following methods do not change their argument, so pass them as const-references.
void SetFont( sf::Font &font );
void SetString( sf::String &string);
In the same way, the getters should return references to const
plus be const-qualified, since they don't modify the instance.
sf::Font& GetFont();
sf::String& GetString();
Provide constructors.It would be nice to initialize the TextArea in the way it will be used, for example with the string, the rect area and the font as arguments. I suggest that the font is defaulted to sf::Font::GetDefaultFont() in case no explicit font is specified. So, one doesn't have to call SetFont() when using SFML's Arial.
Ensure object validity and consistent state.The current default constructor creates an invalid instance. That means that using the public interface (methods GetString() and GetFont()) leads to undefined behaviour because you dereference null pointers. Either you make sure there is always an object with a valid string and font (by not providing a default constructor or setting those attributes to default values). The alternative is to return pointers or throw assertions at failure.
Besides, of the member function SetString() I expect that it adapts the internal text. But this isn't the case. In my opinion, Parse() should be private, as it is an implementation detail. The public interface should be responsible that a new string is automatically parsed and decomposed into the sf::Text chunks. Hereby, the next call to Render() already draws the new text.
By the way, what is the method BlockDebug() for?