Tviberg beat Switzerland's Wendy Holdener (28) in the final of the women's race while Norwegian Thea Louise Stjernesund (26) took the bronze medal ahead of France's Marie Lamure (21).
Schmid was fastest in the men's event, beating Austria's Dominik Raschner (28) with Norway's Timon Haugan (26) taking the bronze.
Norway took three out of the six medals in the parallel events.
Tviberg, who has never been on a World Cup podium before, took a 0.67-second victory over Holdener.
"I’ve never had a (individual) medal before and just to do that is amazing and then a gold on top of that, you can hear my voice is shaking. I always have stuff to say and I am speechless," Tviberg said.
Schmid's win snapped a 34-year drought for Germany who had not had an individual world champion since 1989, when Hans-Jorg Tauscher won the men's downhill title.
"It means a lot - I didn't expect it, and I am so happy to be here with a gold medal, it is just amazing," Schmid said.
"Every run is a new run. You have to keep going. It's no risk, no fun, and that was pretty fun."
The events, in which pairs of racers compete alongside each other over two runs down parallel giant slalom pistes in a knockout format, was held for the second time in the world championships.
Former world champion Marta Bassino (26) from Italy ended in 11th place while her co-winner Katharina Liensberger (25) from Austria did not participate in the finals.
Defending men's champion Mathieu Faivre (31) from France did not qualify for the finals.
The next event is the women's giant slalom on Thursday.
