Volvo Vernensis

I think the most important part in determining the answer is defining the object [Vern's Volvo] in the question. From there it's obvious which entity is the correct choice.

Perhaps the more common definition fits into a context of verbal usage of the term "Vern's Volvo" similar to the way that we might refer to our cars: "Did you hear about how that taxi driver put another dent in Vern's Volvo?" similar to "This morning I followed Sophie's Saab to school." Each of these refers to the vehicle owned and driven by a certain person during a certain instance. Supposing a more specific (accurate) representation, "Did you hear about how that taxi driver made another dent in the Volvo brand car that Vern holds ownership to and drove to work this morning?" which notes that the use of "Vern's Volvo" in the first sentence could refer to multiple sets of Volvo brand parts arranged in the company's proper car form. Vern's Volvo a week after the incident will refer to a slightly different set of parts: one with dent-less panels. Most listeners assume conditions based on the sentence, identifying "Vern's Volvo" as the Volvo vehicle that Vern drives, obviously the one he drove that day when it was dented. With such an assumption, [Vern's Volvo] must be the car at the time of the question, which fits most of our assumptions that [Vern's Volvo] is the Volvo car that Vern owns and drives.

Another possible usage for the term [Vern's Volvo] is as the name for the vehicle (made up of specific parts) that Vern purchased or inherited or something twenty years before this question: his (original car). I imagine it as something like "Herbie the Love Bug," maybe a personalized decal on the Volvo, or even just Vern's name for that specific vehicle; that is, independent of Vern's usage of the (original car) and giving that combination of parts its own label. Assuming [Vern's Volvo] as the name of this (original car), the answer cannot be the car that he is driving at the time of the question, since it does not fit the condition of being made of those specific parts. Note that if the identity becomes a concept, as "Herbie the Love Bug" becomes the name of any talking, specifically painted beetle in a "Herbie" movie, then we return to the first interpretation of context and not the definition as a certain set of parts.

Assuming that [Vern's Volvo] refers to the (original car) alone, the answer to the question may be the pile of parts. The nature of the question is determined by the one asking it, so if the set of parts, perhaps more accurately referred to as "the parts formerly composing Vern's original Volvo," satisfies his or her query then the pile is the answer. If the original conditions for [Vern's Volvo] include not only that it is made up of specific parts, but also that they are fashioned in a specific manner into a car, then neither is the answer.

I, making the assumption that [Vern's Volvo] refers to the Volvo brand car that Vern drives, would say that it is the car that he considers his at the time the question is asked, that is, the assembled car. This is because I assume that the question is asking for the car he currently drives because the question is likely asking with the hope that the answer would, in the future, again be identified as [Vern's Volvo], as the car that Vern currently drives would likely be. If I knew that the question was asked by the dealer that sold Vern the car twenty years ago and has always loved the alliteration and referred to that car that he sold as "Vern's Volvo," then I might assume his definition of [Vern's Volvo] refers to specific parts and identify the pile in Grace's Garage, giving this friend the condition that the car is in a fragmented form, should that violate his definition of what constitutes [Vern's Volvo].