Younghoe Koo missed a 44-yard field goal attempt with 5 seconds left after Philip Rivers maneuvered the Chargers (0-2) into scoring range in the final minute before a crowd of 25,381 in their first home game since relocating to the Los Angeles area after 56 seasons in San Diego.
The Chargers' Antonio Gates caught his 112th touchdown pass as a tight end, breaking his tie with Tony Gonzalez for the NFL record.
The 37-year-old Gates caught a 7-yard pass from Philip Rivers midway through the third quarter to surpass Gonzalez's mark.
The milestone is another remarkable achievement for the 37-year-old Gates, who didn't play college football and entered the NFL as an undrafted free agent. He teamed up with Rivers on the 85th TD pass of their parallel careers, but their first in Los Angeles after starting in San Diego.
Gates also made his 900th reception in the first half, becoming the third tight end in NFL history to hit the mark.
Jay Cutler passed for 230 yards and Jay Ajayi rushed for 122 in the delayed season opener for the Dolphins, who spent the past nine days in California after leaving South Florida early to avoid Hurricane Irma. Parkey did most of their scoring, connecting from 30, 28 and 35 yards before burying his longest kick in the waning moments.
Rivers passed for 331 yards.
Rivers and the Chargers moved 54 yards in 52 seconds to set up Koo, whose potential game-tying field goal was blocked in Denver last week. The undrafted rookie kicker pushed this one barely right.
Kenny Stills caught a 29-yard TD pass in the third quarter from Cutler, who went 24 of 33 without an interception in his Dolphins debut.
Parkey's 35-yard field goal trimmed Los Angeles' lead to one point with 6:28 to play, and Miami's defense held. The Dolphins got the ball back with 3:02 left, and Cutler got them into position for another big kick from Parkey, the Jupiter, Florida, native claimed off waivers two weeks ago.
The Dolphins were well-rested after their home opener against Tampa Bay was postponed, leaving them to play their latest season opener since 1977.
The Chargers' latest last-minute loss dampened the festivities around a fresh start for the franchise, which brought five members of its inaugural 1960 Los Angeles Chargers to its temporary home at 27,000-seat StubHub Center, which was packed with a lively mix of both opponents' fans.
Melvin Gordon scored the Chargers' first touchdown in their new home from 1 yard in the second quarter to cap a 70-yard drive, but both teams struggled to put together sustained scoring marches.
Next up for the Chargers is the continuation of a three-game homestand with a visit from the Kansas City Chiefs, who have beaten their AFC West rivals six straight times.