Since there is no verb conjugation in Mandarin Chinese, how should one
know that a verb in a sentence is in passive voice?
In the strict sense, Chinese verbs have no passive voice; nor do they have active voice, middle voice, tense, moods, infinitives, number, or person. In other words, Chinese verbs are not systematically organized along these parameters in any meaningful way.
In the larger sense, Chinese does have means of addressing the semantics and pragmatics associated with the English passive voice; however, there is no one-to-one equivalence, since this is not a concept in Chinese at the systemic level.
Before understanding how Chinese operates, it is good to understand that even in English the concept of passive voice is more limited than it might seem, since English verbal nouns and infinitives often do not show a clear difference in form. To see this lack of distinction, consider the phrase: "I didn't approve of your arrest." It is ambiguous if this phrase means: "I didn't approve of your being arrested" or "I didn't approve of your performing the arrest." In context, English speakers manage fine with such ambiguity.
Also, consider the sentence: "That was hard to hear." In my view, "to hear" feels like it has a passive meaning. An alternative view would to view "to hear" as a reduction of a longer phrase, such as in the sentence "That was hard for us to hear." At the system level, English does not require the voice of "to hear" to be clarified if you delete the agent.
Now consider your Chinese examples:
You can consider this the equivalent of the odd sentence "Your parcel is not at our office for the taking." In English, the form of "taking" is active, but the meaning within the sentence is passive. This grammar is like "That parcel is not easy to take." In both English and Chinese, these sentences express an orientation toward a passive object that makes the "passive" semantics quite natural, even if some English constructions would require an explicitly passive verb (e.g., "This parcel cannot be taken from our office." This choice between forms is an aspect of English grammar, but not Chinese grammar, which has only one form.
This sentence is the equivalent of the odd sentence "With respect to your library card, there have been successful arrangements." Neither English nor Chinese expresses an agent, yet neither includes a passive construction. Of course, more natural English might use a passive construction. Natural Chinese does not have such an option; instead, it has a topic-comment structure not available to natural English.
Think of this as the equivalent of: "My friend's/friends' child does not need for her parents to provide care." In Chinese, the phrase 不用 usually takes a verb or verb phrase as a complement and is oriented toward the point of view of what lacks the need. In this case, we need to interpret the needs from the point of view of the child and so including the words 她爸爸妈妈 before 管 just expresses the agent providing the care, and there is no need to provide a pronoun after 管 in this type of construction, just as there is none in the English equivalent I provided. If the parents were the object of the caring, these words would come after 管 in the normal complement position.
Compare your sentence with 这些衣服很干净，不用洗 ("The clothes are clean and don't need washing"). Even in English, we would not say "*The clothes are clean and don't need being washed" but can use either "don't need to be washed" (with a passive form) or "don't need washing" (without a passive form), since we do not systematically mark voice in gerunds.
The most difficult example, 他想把刚买的苹果带来.
Which translation is correct?
a) He wants to bring the apples he has just bought. (active)
b) He wants to bring the apples which/[that] are/[were] just bought. (passive)
The Chinese does not specify who bought the apples, so either translation is possible, depending on context. The Chinese sentence would be said only if it were obvious or unimportant who did the buying and so even if it is technically ambiguous, it would not usually be pragmatically ambiguous. Most typically, if an agent is indicated, as in this case, but unspecified, you will assume that the most recent agent used as a topic is the agent. That process would yield the first translation as the more likely one, where you state that "he bought the apples."
Be mindful that the two English sentences carry their own pragmatics and so introduce distinctions and nuances not inherently present in the Chinese sentence. At the same time, the Chinese sentence could be said in other ways that would introduce their own particular pragmatics, distinctions, and nuances.