You can parse the string pretty simply yourself:
// Extract string up to position of the ,
Temp := Copy(Temp, 1, Pos(',', Temp) - 1);
// Remove double-quotes from resulting string
Temp := StringReplace(temp, '"', '', [rfReplaceAll]);
Here's a sample console app that uses the logic above on the sample response you provided, and outputs the final contents to the console:
program Project2;
{$APPTYPE CONSOLE}
uses
SysUtils;
const
LinkStr = '"http://core2.staticworld.net/images/article/2014/01/counterfeit_android_apps1-100227383-medium.jpg","http://core2.staticworld.net/images/article/2014/01/counterfeit_android_apps1-100227383-medium.jpg"';
var
Temp: string;
begin
Temp := Copy(LinkStr, 1, Pos(',', LinkStr) - 1);
Temp := StringReplace(Temp, '"', '', [rfReplaceAll]);
Writeln('After: ' + Temp);
ReadLn;
end.
Output: