How do I properly wire GFCI outlets in parallel?
GFCI receptacles have two sets of contacts, line, and load.  The Line side of the receptacle is used to power the device, while the load side is used to power other devices down the line.  Any device connected to the load side of a GFCI receptacle, will be protected by the GFCI receptacle.

For example, if you have a setup like this (which I assume you have).



There is no need to have a GFCI receptacle as the second receptacle, since it will already be protected by the first GFCI receptacle. Because of this, if the first device trips all devices on the load side will not be powered (as you have noticed).

You can use pigtails to connect the receptacles like this.



But in a setup like this, you'll be required to have a GFCI receptacle at both outlets. The devices down stream are no longer protected by the first GFCI receptacle, because they are not fed by the load side of the device.

FYI:
This is what it would look like if the receptacles were wired in series.