If we look at the dictionary definitions they are almost synonymous. Especially the applicable definitions of field and realm read nearly identical and realm even uses field in its own definition.
In terms of is defined a little different in wording but still reads very similar.
field 2: A particular branch of study or sphere of activity or interest. - ODO
realm 1.1: A field or domain of activity or interest. - ODO
in terms of (or in —— terms): With regard to the particular aspect or subject specified. - ODO
If we look at the following ngrams we can see that in terms of is hugely more used than the other two variants.
While the general dominance of in terms of cannot be established with ngrams for the two fields security and defense, the usage of realm(s) of) is always used fewer in the digitized books. Thus, a clear preference for any of the wordings cannot be established. If you want to use a parallelism, I see nothing that precludes you from doing so.
Lastly, note that the sentence is the transcription and translation of something Angela Merkel said. It is not unusual for politicians to add another train of thought after they were almost finished with a sentence. As such the BBC couldn't contract the parallelism without taking some freedom in the translation. However, I see no changing in meaning if they had. Nothing other than making Merkel seem a bit more eloquent at least.
I tracked down the original video. It's on the Facebook page of the Bundesregierung.
Es geht jetzt nicht darum, einfach von einem Gipfel die Lösung der Probleme Europas zu erwarten. Wir sind in einer kritischen Situation. Sondern es geht darum, durch Taten zu zeigen, dass wir besser werden können: im Bereich der Sicherheit - innerer und äußerer Sicherheit - , [bei der] Bekämpfung von Terrorismus und der Zusammenarbeit im Verteidigungsbereich, dass wir besser werden können in den Fragen von Wachstum und Arbeitsplätzen.
-Angela Merkel, 16.09.2016
She didn't say what's in the brackets, that was inserted by the subtitling on the video. I emphasized her usage of Bereich, which is explicitly the same in im Bereich der Sicherheit ... (area of security) and Zusammenarbeit im Verteidigungsbereich (cooperation in the area of defense). The last is part of a compound word but there's no difference between Verteidigungsbereich and Bereich der Verteidigung. Generally it's a (very) loose and shortened translation. They even skipped one and a half sentences. Anyways, the use of different words cannot be traced back to the original.