Park struck first midway through the first half and Koo sealed victory late on for Thomas Tuchel's men, who are now breathing down the necks of sixth-placed Wolfsburg.
Freiburg, who started 2014 with a 3-2 win over Bayer Leverkusen, had no reply as they were once again shown the severity of the situation they find themselves in.
Mainz made a flying start to the game and thought they had gone in front when Shinji Okazaki diverted a cross into the top right-hand corner but after lengthy deliberation with his linesman, the referee ruled the goal out.
Mainz did not stop attacking, though, and Okazaki continued his quest to break into double figures this season against a Freiburg side who, initially at least, looked a mere shadow of the one which had beaten Leverkusen a week ago.
It was not too long before Mainz took the lead, with Park trying his luck from nearly 30 yards out. The ball took a deflection off Immanuel Hohn, looping it over a forlorn Oliver Baumann in the Freiburg goal.
Chances at the other end of the field could be counted on one finger, and it was no surprise that Freiburg coach Christian Streich made alterations at the start of the second half with Francis Coquelin and Felix Klaus both coming on.
They seemed to make a difference too, with Admir Mehmedi finally getting in a notable shot.
But it was another substitution, one for Mainz, which made the difference. Koo came on for Yunus Malli and, in only his second game for the club, he scored his first goal.
The former Augsburg and Wolfsburg midfielder cut in from the right and sent a fine curling effort with his left foot into the far corner, beyond Baumann's outstretched hand, to seal victory number two of 2014 for Mainz.