Lewis and Short only give present stem forms of the verb appărĕre, appărio. They say, quite rightly so, that it comes from ad+părĕre, and one would therefore expect the conjugation to be as that for părĕre. I interpret the lack of perfect forms in the dictionary entry to indicate that those forms are not attested, not that they are illegal in any sense. If I want to use the perfect forms of this verb, how should I go about it?
The conjugation of parere is pario, peperi, partus/paritus. Therefore one natural guess is appario, appeperi, appartus/apparitus. However, it is not unusual for reduplication to be lost after a prefix, as in cucurri > concurri. Therefore also appario, apperi, appartus/apparitus would make sense. Which one should I pick, appeperi or apperi, or something else? Or are both fine?
If there is a general rule that allows to decide whether reduplication is retained with prefixes, that would certainly settle this question. If no such thing exists in the literature, then one can only presumably argue by analogy to examples.
The verb in question is not to be confused with appārēre, from which we derive e.g. appāritio. I am not aware of any such derivatives of appărĕre. There are three verbs pario in Latin, but they are distinguished by their infinitives: pariare, parere, and parire. The verb of this question is of the third conjugation, but it is somewhat confusing that there are so similar-looking verbs in all other conjugations.