Spanish-based HRT have not scored a point in nearly three seasons in Formula One and usually start at the back of the starting grid.
Schumacher, who has announced his retirement at the end of the season, started the last race in Japan alongside Karthikeyan at the back of the grid after collecting a 10 place penalty for causing a collision at the previous race in Singapore.
De la Rosa was testing a new rear wing for HRT in the afternoon session as well as tyre options.
Schumacher has now been reprimanded twice this season, the previous one coming at the Spanish Grand Prix in May when he held up McLaren's Lewis Hamilton - his replacement at Mercedes next year - in practice.
The German lapped fifth fastest in the afternoon session in South Korea.
Red Bull laid down an ominous marker as they claimed the top two spots in second practice.
The Milton Keynes-based team were the dominant force at last weekend's Japanese Grand Prix and they look like again being the pacesetters in Yeongam.
Sebastian Vettel, just four points behind Fernando Alonso in the championship standings, set a time of one minute 38.832 seconds to beat team-mate Mark Webber to top spot by just three-hundredths of a second.
Third place went to Alonso as Ferrari showed signs that updates to the F2012 will make them a more competitive proposition here than in recent races, although he was 0.3 seconds adrift of the Red Bulls.
Jenson Button was fourth in the first of the McLarens.