The spell is usually left up to the GM but some commands, such as "Grovel!" have specific, predetermined effects. "Grovel!" does apply the Prone condition; "Halt!" does not apply the Paralyzed condition (or any condition)
Much of the spell is left up to the GM's interpretation:
You might issue a command other than one described here. If you do so, the GM determines how the target behaves.
That said, the "Grovel!" command in particular is an example the spell provides:
Grovel: The target falls prone and then ends its turn.
If you fall prone you are prone, this means you are affected by the prone condition. This is shown because everything that makes a target prone uses that wording or similar (such as knocking a target prone).
Similarly, effects do not say "you apply the blinded condition to your target" but things like "you blind the target" which, through standard English, is an equivalent statement.
The "Halt!" command states:
Halt: The target doesn't move and takes no actions...
All it does is cause a target to not move and be unable to take actions, nothing else. No conditions are applied because spells do only what they say.
Note, that being "unable to take actions" definitely also prevents bonus actions as the rules state (PHB page 189):
Anything that deprives you of your ability to take actions also prevents you from taking a bonus action...
But whether this also prevents reactions is not entirely clear and is discussed in the question "On the turn after Haste ends, can you use a reaction?"