Our preferences draw arbitrary distinctions when, and because, what we prefer is in no way preferable. It is arbitrary to prefer one of two things if there are no facts about these things that give us any reason to have this preference. [On What Matters, Chp 3.]
In order to explain why basing a meta-hedonic preference on the day of the week is arbitrary, whereas basing it on the felt quality of the experience is not arbitrary, Parfit thinks we must appeal to the substantive normative fact that only differences of the latter sort are reason-giving. I think this is not quite right.
We can specify a non-normative sense of 'arbitrariness' in terms of how differences in one's preferences match up with differences in natural properties. Since "being Tuesday" is a merely conventional (rather than natural) property, there's a straightforward sense in which the Future Tuesday Indifferent agent is being arbitrary: they are not treating (naturally) like cases (normatively) alike.
Still, I don't think this non-normative sense of arbitrariness is enough to get the proceduralist out of trouble. For we can imagine a variation on the case where, instead of their preferences featuring Tuesday (as such), the agent's degree of meta-hedonic concern varies according to the waxing and waning of the moon (or some similarly natural variation). Suppose that they have full ordinary concern about future pains experienced when the moon is full, but only half as much concern for agony felt under a half-moon, and no concern at all to avoid agony felt under a new moon (regardless of whether they'd recognize it as such at the time). In this case, there doesn't seem to be any straightforwardly procedural grounds for criticizing the agent -- their preferences are not arbitrary in the non-normative sense -- but it still seems objectively crazy.