Are context free grammars the only ones that have parsing trees?
If you look at the Chomsky Hierarchy and the corresponding definitions
of grammars for each of the four types you can make the following
remarks:


there does not seem to be any natural concept of derivation
(generation) tree for type 0 grammars, the unrestricted kind, that
would apply to all.
for context-sensitive (type 1) grammars, at least in the form given
in the wikipedia article, one could associate a tree structure to
the derivation of the string, but this tree, even annotated with
the grammar rule applied at each node, would not be sufficient to
recover simply the order of application of the rules. You would
have to actually associate some partial order with the tree nodes.
I do not know whether this was ever used.
The context-free case (type 2) is well known.
The regular case (type 3) is well known too. It may be seen as a
special case of CF. But regular grammars, and associated tree
structures are not interesting since they are trivial. They do not
really provide useful generative information, nor do they impart
structure on the generated string. Note however that CF grammars
with interesting tree structures could well generate regular
languages.


However, this is not the end of the story. There are many kinds of
grammars, between the regular family and the full context sensitive
(CS) family, i.e. around the CF realm. Many can associate a derivation
tree with the generation process, and they can also associate a
derived tree with the generated string, thus giving it a structure.

These two trees are the same in the case of CF grammars, but, as I
indicated in my answer to another of your questions, they may also
differ, notably in grammars involving multi-parts constituents. such
as Tree Adjoining Grammars (sometimes called mildly context
sensitives) or more generally Linear Context-Free Rewriting Systems
(LCFRS). These grammars depart somewhat from the basic model of
Chomsky hierarchy, as do many others, such as indexed grammars.

For all these formalisms, the languages defined are usually in the context
sensitive realm, and cannot be generated by a CF grammar. The derived
trees associated with the strings are thus also beyond what can be
done with CF grammars. However, many of the formalisms considered in
the literature have sets of derivation trees that are the same as those
of Context-Free languages.

The set of parse trees (meaning here both derivation and derived
trees) for context-free language is a tree language. The family of
parse-tree languages associated with CF grammar is precisely what is
called regular sets of trees. They can be defined by Regular Tree
Grammars (that may look suspiciously like string CF grammars). They can be
recognized by finite state tree automata.

What distinguishes CF Grammars is that their derived trees are regular
sets of trees, which is not the case for the other formalisms. However,
many of these formalisms have regular sets of derivation trees, the
same as CF languages.