Let $X$ be a compact metric space. Say that two compact subsets $E,F\subset X$ are *parallel* if
$$ dist(x,F) = dist(y,E)$$
for all $x\in E$ and $y\in F$. Here $ dist(y,E) = \inf\{d(y,z):z\in E\}.$

The overall question I would like to understand is the following:

Let $X$ be a compact (Hausdorff, second countable, hence metrizable) topological space with a collection of disjoint compact subsets $\{E_t\}_{t\in T}$. When is there a metric on $X$ that makes every pair of sets in $\{E_t\}$ parallel?

I generally think of $T$ as uncountable and $\{E_t\}$ as a partition of $X$, but I don't see the need to assume this.

There is at least one clear necessary condition that $\{E_t\}$ must satisfy: Namely, if $\{x_n\in E_{t_n}\}$ is a sequence converging to $x\in E_t$, then every element $x'\in E_t$ is the limit of a sequence $\{x'_n\in E_{t_n}\}$, **and** every sequence $\{x''_n\in E_{t_n}\}$ accumulates only at points in $E_t$.

Call this condition **(*)**. (The part after the **and** in condition (*) was added after MTyson's example.)

For example, if $X=[0,1]$ and $\{E_t\}$ consists of $[0,1/2]$ as well as all the singletons $\{x\}$ for each $x\in (1/2,1]$, the collection fails condition (*) and can't be made parallel.

So I refine the question:

Let $X$ be a compact (Hausdorff, second countable, hence metrizable) topological space with a collection of disjoint compact subsets $\{E_t\}_{t\in T}$, satisfying (*). Is there a metric on $X$ that makes every pair of sets in $\{E_t\}$ parallel?

As I don't know if I believe this, I also ask:

Are there conditions, in addition to or instead of (*), that ensure the sets $\{E_t\}$ can be made parallel by the appropriate choice of a metric on $X$?

I'd be interested in counterexamples, other possible obstructions, references to similar things in the literature, or general sufficient conditions (even if they assume more about $X$ or $\{E_t\}$).

One could imagine cases in which $T$ is a topological space and $t\mapsto E_t$ is continuous in some topology on the space of compacta of $X$ (e.g., that induced by the Hausdorff metric coming from some metric on $X$.) Maybe this is equivalent to my formulation. Regardless, if that makes the problem easier or more well-known then by all means assume it.