Yes; casting is not communication
Many if not most spell text blocks follow the pattern of first description, then proscription. That is, first a narrative expression of the spell, then what effect it has on the rules. For a symbol of discord, the description is "a target bickers and argues with other creatures", while the proscription is "it is incapable of meaningful communication and has disadvantage on attack rolls and ability checks".
Discord penalizes attacks and ability checks, but no mention is made of its effects on spells (in contrast to, for example, the Silence spell). I am taking for granted that verbal spell-casting is not affected, specifically because the principle of "spells do what they say they do"; it does not say that verbal spellcasting is affected, so it is not. I am attempting to justify this post-hoc, rather than arguing whether verbal spellcasting should be allowed, as other answers may be.
So, if the target is prevented from meaningful communication, why is spellcasting permitted? Because spellcasting is not communication. Communication has an intelligent recipient. The spell says that the target bickers and argues with other creatures. Verbal spellcasting remains functional because it does not have an intended listener, it does not have a creature as its recipient. The caster is still fully able to use verbal components (cf.), to "produce particular combination of sounds, with specific pitch and resonance". What they are now unable to do is produce an intelligible message for a specific listener, including modulating that message - facing them, enunciating, gesturing, coding, and all the other components of communication.
For a deeper narrative description of what this is like, consider the eponymous Discord herself, the Greek goddess Eris. She causes strife by setting people at odds, to 'bicker and argue' amongst themselves. She foments interpersonal conflict, not impediments to speech.
One interesting effect of this interpretation is on spells that grant control over other creatures. A creature under the effect of Discord would still be able to cast suggestion, command, or similar spells - but their target would be unable to understand the instructions given them, due to the power of the Discord to prohibit meaningful communication.
As an interesting example of how this phenomenon might work, albeit in the listener, and not the speaker, consider Songs of Praise and Songs of Praise 2. In these videos there is unaltered sound of choral music but with incorrect subtitles. Watching the subtitles predisposes you to hear the sounds differently than they are actually produced - the sounds themselves (required for V casting) are unchanged, while the meaning is incomprehensible because the part of your brain that creates meaning from sound has been hacked.