Pulgasari
While Pulgasari was made decades before Tristar's remake, the movie was shelved for ages due to its political messages that weren't appreciated in its home country of communist North Korea. When the movie finally saw the light of day again it used a red glow around the title which kind of resembles Godzilla's. Whether or not that was the intent may never be known. Notable is the involvement of Toho crew members in creating the iron creature.
Gargantua
Although the plot and creature design differ greatly from either the remake's style or the original Godzilla series' formula, Gargantua was made by FOX to benefit from the hype. The box cover seems to draw from the remake's most famous poster of Godzilla's eye.
Reptilian
The South Korean film Yonggary was arguably made in answer to the popularity of Tristar's Godzilla. The U.S.A. DVD cover, however, undeniably copied Tristar's signature "Green Godzilla Glow" for its Western release as "Reptilian."
"Remote Control Godzilla"
Also pertaining to "Yonggary" is this R/C controlled Yonggary, bootlegged under the name Godzilla. The "De-motivational" frames and text were added later.
MAD #370
The ever relevant MAD magazine used the remade Godzilla's likeness for issue 370's front cover, baring the hilarious slogan "Godzilla bites!"
Star Wars Episode 1: The Phantom Menace
Not a parody within Episode One itself, but rather it's marketing. The official webpage duplicated the "Green Godzilla Glow" logo, and parodied the slogan of the movie, "Size Does Matter," with "Plot Does Matter." Ironically, Episode One was later criticized for a lack of plot.
Anaconda (Inversion)
Anaconda's release actually came before Godzilla, though their style of showing only the eyes is similar. It's possible the Godzilla marketing team took note, and appreciated the approach.
Komodo, Curse of the Komodo, Supercroc
Komodo likely used the "eye cover" from the company's past success with Anaconda. It's also likely that Curse of the Komodo employed a similar cover design so that consumers might mistake it as Komodo, which was infinitely better. The angle at which the eye is present is closer to Godzilla, though and the addition of the helicopter could be to coax Godzilla fans into purchasing the movie. No, the creature doesn't actually fight an attack helicopter in the movie. 2007's Supercroc use of the eye close-up is curious. As a side note, it's the worst movie of the three, and possibly the worst movie mentioned on this page.