In the absence of any other information, if your PC is looking for where to store files or, as here, looking for where to load them from, it will normally open the last folder which you accessed. If for some reason it can't do this (e.g. if the folder has been deleted) it will always go to My Computer. It could be that there's some sort of registry error preventing your PC from following the normal pattern. I could suggest some possible registry fixes but, to be honest, it's probably not worth the hassle. After all, it only takes a few seconds to navigate from My Computer to My Pictures.
All the same, if you want to try to persuade your PC to use the correct default, just try saving any picture from the web to My Pictures. Because you've accessed this folder, your PC ought to treat this as the default folder. (To check, without even going to another page, right-click the AnswerBank logo at the top-left of this page. Select 'Save as'. Then navigate to My Pictures and save the logo there. If you then go to tinypic.com and click browse, your PC should now use My Pictures as the default folder).
Chris