What is parts only warranty does my laptop battery and other parts still on warranty or not?

First of all, don't be surprised if a forum moderator deletes your image as it includes the service tag number of your system and for some reason that makes them have a cow.  

Looking over your summary it does indeed say you are currently under a parts only warranty until August 2016.

A parts only warranty means Dell technical support will accept a call from you and troubleshoot problems with you and if they determine that a replacement part is necessary they will ship it to you for you to install (or have installed by someone at your expense).  The good news is that parts dispatched usually come with a confirmation email that has a link to installation instructions.  So there is a good chance you will be able to install the part yourself.  Also, they may or may not require that you return the defective part.  Carefully read the paper work included with the replacement part or you may later receive an invoice for the replacement part if you are required to return the defective part but failed to do so.

As noted in the fine print at the bottom of the image you provided, batteries are not necessarily covered by the warranty.  Dell only warranties batteries for the first year.  After that they are not covered by the system warranty.  You are in your third year of warranty for this system so unless you bought a new battery from Dell in the last year then the battery is out of warranty.

Generally extended warranties do not make sense (are not a good value) except on expensive systems.  Most failures are the cause of some accident (liquid spill, unit dropped) and few of Dell warranties cover 'accidental damage.'

The most common repairs for laptops are flat panels replacements, disk drive replacements, and keyboard replacements.  Each of these parts are well under $75 and are all fairly easy to install by a handy person.  The hardest would be a disk replacement because it involves hours to reload the operating system, updates, drivers, your software, and your data... Often people just buy a new system rather than make the effort to rebuild all the software in the event of a hard disk failure.   Also most major credit cards extend the factory warranty for an additional year making a one year Dell warranty as good as two.

Hope that you find this to be clear and helpful to you.