Parfit allows, in Chapter 1 of On What Matters, for a distinction between different kinds of way in which things can be termed "good" though, as Smith points out, it is unclear whether these are three types of thing that share the common property of "goodness" or, rather, whether "goodness" is itself something that can be determined as existing in three distinct ways. This topic, whilst of some importance in relation to Parfit's response to Sidgwick, is not one that I will attempt to adjudicate on in this posting.
However, the types of goodness that Parfit considers are all related in being forms of goodness that are termed by him "reason-implying". Something that is "good" in a reason-implying sense, is something that has "certain kinds of fact" about it that, in at least certain situations, give us strong reasons to choose it (or use it or produce it etc). This is what Parfit terms "the most important use" of goodness. Since the types of reason-implying goodness refer, in some way, to "facts", the next question concerns what might be meant here by a "fact" and what ways the "facts" in question could give us reasons.
Since we are concerned, when Parfit refers to "reason-implying" senses of goodness, with reasons in the sense of reasons for action, it would be expected that the types of goodness that Parfit is concerned with would be good in the sense of providing us with a strong reason to act in certain ways. There are three such types of goodness discussed by Parfit. The first is that something is "good for us", in the sense that it contributes, in some way, to our well-being. However, Parfit does not intend well-being to be considered here in a purely instrumental way, as when something contributes to my "well-being" by improving my finances. Rather, Parfit is focusing on "well-being" in an intrinsic sense that involves "one of the features of our lives in which our well-being consists" or what he elsewhere (in Reasons and Persons) describes as something that enables our life to go best. Now this seems to involve reference to what Rawls termed "primary goods" and can include states of certain sorts but broadly speaking these types of goods are things we have self-interest in relation to without being selfish.
This first type of goodness, goodness-for, is understood by Smith plausibly to involve phenomenological considerations and does at least partially get captured by hedonic theories of goodness. By contrast to such goodness-for there is also person-relative forms of goodness in which we have reasons to be partial towards people who we are related to in certain ways. This person-relative goodness is something that has raised considerable questions in contemporary moral philosophy since some take it to require departures from formal universal procedures (as Bernard Williams famously argued in his "one thought too many" discussion).
The third type of goodness is goodness in an impartial sense though this kind of goodness is determined in two different ways by Parfit. On the one hand, there is impartial goodness in terms of reasons to care about the well-being of anyone and, on the other, there are the reasons we would have if we had an impartial point of view. The first type of impartial reason is claimed by Parfit to involve substantive beliefs about well-being whilst the point of view of impartiality need not give us such beliefs automatically since we might have to over-rule as well other types of preferences (such as aesthetic ones though, oddly, Parfit views these as not involving "reasons" at all but perhaps he only means reasons that should normatively matter). When something is impartially good it is impersonal since we don't prefer it because it would be good-for someone even though such benefits are what, in the outcome, would be termed "good".
Parfit's consideration of "objective" theories of value suggests that such theories have their reasons provided to them by "facts about the objects" that are involved in reasons and aims. These contrast with facts that are, apparently, merely facts about "us" with the latter being stated to be the concern of "subjective" theories of value. So an "objective" theory concerns itself not merely with aims and desires but with the worthwhile character of them in relation to their "objects". Given that the theory that Parfit is elaborating and terming "objective" is nonetheless concerned with desires and aims and with the way in which such desires and aims are formed in a sense that is worthwhile Smith claims that Parfit's "objective" theory is only a form of desire-based theory. This is because, as was demonstrated in the last posting, there is no problem with a "subjective" theory adopting the view that there is required reflection on desires and principles needed for their formation. Since Parfit also takes Kantian theories of normativity to be "subjective" in these ways it is clear that Smith is right about this.
So, just as in the last posting it emerged that Parfit's distinction between subjective and objective theories of reasons had no valid place for Kantian theories of normativity so, in this posting, we can see that Parfit's classifications of types of goodness does not meaningfully allow him to distinguish his account of reason-implication from a Kantian one. In neither of these ways can Parfit sustain his distinction between "objective" and "subjective" theories of reasons or his apparent insistence that his own theory of reasons is one that is intrinsically different from and opposed to Kantian theories of reasons.
Smith's general claim is that Parfit's own theory is a desire-based one that depends on the principle that "if someone believes that p and that p produces q, then they believe that q is something intrinsically desirable". If Parfit's account does involve commitment to any principle of this sort (and the three forms of goodness can all be converted to reference this) then it is the case that he has failed to meaningfully define his theory as "objective" and hence distinct from the "subjective" theories he is opposed to.