The gemara in Pesachim 76b forbids one from cooking meat and fish together because of danger.
But can one cook meat and fish dishes separately at the same time in the same oven? There can be a need to do this when preparing lots of food at the same time, e.g., before a 2-day holiday.
Many halachic decisors permit cooking meat and fish dishes simultaneously provided at least one of them is covered (SA YD 108:1). After the fact, it is permitted to eat the food even if not covered (Rema on SA YD 116:2).
DailyHalacha writes
When meat and fish are cooked together in a closed area, the steam produced by one is absorbed by the other. Therefore, one should avoid cooking meat and fish together in an oven or closed grill, unless the fish or the meat is covered, such as with aluminum foil and the like, so that it is shielded from the steam produced by the other. Nevertheless, after the fact, if one did cook fish and meat together in an oven without covering one of them, they are permissible for consumption.
dinonline brings the following
Although the Gemara (Pesachim 76b) does imply that it would a problem, even bedieved, to cook meat and fish together in the same oven, the fact that this halachah is not ruled by Rif or Rambam has led poskim to be lenient. Although we continue to be careful not to eat fish and meat together, many authorities are lenient concerning fish or meat that absorbed the taste (alone) of the other.
For lenient opinions, see: Issur Ve’heter 39:25, as cited in Rema, Yoreh De’ah 116; Shach, Yoreh De’ah 116:1; Taz, YD 95:3; Darchei Teshuvah 116:16 (quoting from several authorities); Ha’elef Lecha Shlomo 312. For more stringent opinions, see Peri Chadash (116:4); Peri Toar (116:3); Chochmas Adam (68:1).
Lechatchilah, one should not cook meat and fish [uncovered] in the same oven, but bedieved, one may rely on the lenient opinions.
R Yitzhak Yaacov Fuchs writes (in his book Hakashrut, p. 418)
Lchatchilah, it is forbidden to put meat and fish together in the same baking compartment in the oven even though each one is in a separate baking utensil. It would be permissible if one of the baking utensils was covered entirely in a complete manner (wrapped all around and hermetically closed). Bdieved [after the fact] if not fat or juices dropped from one utensil to the other it is permissible to eat both foods (Aruch Hashulchan YD 116:10)
R Ari Enkin summarizes
This would logically extend to restrict baking dishes with these two types of foods in the oven at the same time, unless both pots are well covered, lest any tastes, via the steam, spread from one to the other.
See also here at length from Eretz Hemda which concludes
while it is halachically safer to not roast meat and fish uncovered in the same oven, it seems reasonable to do so in a regular, large oven when there is a need
Kitzer Shulchan Aruch Siman 33:1
One should not roast meat together with fish in one oven if the oven is small, unless one of them is covered, [for otherwise the aroma of one gives flavor to the other.]
See Rama, Yoreh Deah 116:2 (see also Taz and Shach there)
But in out ovens that are large, this is no need to be concerned.
The footnote here says that the author is speaking of ovens in his day and a rabbinic authority should be consulted to be determine whether one's oven is large in this respect.
In conclusion, one of the dishes needs to be covered, unless according to a rabbinic authority, your oven is large.
All other answers offered, take the tact that it is a granted that it is contemporaneously forbidden to eat meat and fish together. It is worth noting that this assumption is not universal.
There are today Ashkenazim and Sefaradim that understand there to be no prohibition extant and therefore do not refrain from eating meat/fish. And all the more so would certainly not refrain from cooking them in the same oven.
For a great analysis of this issue, see R. Zekharyah Zaramati's responsum here.
