As @user13484 pointed out, the general concept of two (or more!)
melodies sounding together is counterpoint ("note against note").
The term counterpoint is used when the melodies that are sounding simultaneously are independent and are more or less of equal importance.
Both rhythm, pitch but also timbre can be used to achieve independence.
The term is about a technique or texture, and as such it may be applied to a section, or a movement or even an entire composition.
"Independence" is not a matter of all or nothing.
Although the melodies might have a different rhythm, a melody might have the effect of emphasizing only particular accents of a main melody.
Or although a melody might be playing different pitches, it might do so in an identical rhythm.
So in those cases, the extra melody is not truly independent.
With regard to this piece, the parts that are entering at 0:15 do not really seem to carry a new or separate melody.
The new part serves mainly to give the piece a new rhythmic pulse.
The head of the new part serves to accentuate and repeats the same pitch a few times.
The tail of the phrase basically mimics the main melody (it's not exactly parallel since the rhythm is slightly different, but the pitches are).
So there is a certain lack of independence, and I'd argue the parts also aren't equally important, which is why I would not call this true counterpoint.
I'd call this simply "accompaniment".
At 1:30 though, something starts to evolve that does sound like true counterpoint, first in 2 parts, and then at 1:45 in at least three.
Here you can clearly here different parts that are independent in movement, curve, rhythm and pitch.
music.stackexchange.com/users/8883/Roland Bouman
Formally, it would be counterpoint, but since the music is not as a whole constructed in that manner, I'd just call it a countermelody.
music.stackexchange.com/users/13484/user13484
Quodlibet ( Latin: “what you will”) musical composition in which several well-known melodies are combined, either simultaneously or, less frequently, sequentially, for humorous effect.
Quodlibet can also refer to an amalgamation of different song texts in a vocal composition.
While simultaneous combinations of two or more melodies go back to the 13th century (motets using, for example, a chant melody and a secular tune), quodlibets were especially popular in the 15th and 16th centuries.
In Germany numerous instances are found in manuscript collections of polyphonic (multipart) songs.
Britannica.com:
Encyclopedia article about "quodlibet"
music.stackexchange.com/users/21872/Tanti
I think Mingus himself called this structure of melodies stacked on top of each other - in many of his pieces there are more than two, e.g. in Wednesday Night Prayer Meeting - a "pyramid".
music.stackexchange.com/users/10031/Jan Johannsen
In a general-usage sense, the word "counterpoint" could serve, though the word counterpoint names a very specific technical form that follows certain rules of construction.
Because of that, the two lines in Fables of Faubus fail to be "counterpoint" but are most definitely "countermelody."
They could be called "contrapuntal lines" without raising the "counterpoint pedant" alarm. :)
I think the word "mashup" is a horrible way to describe/think of the concept.
There's a big difference between two melodies that were written to work together, vs a pair of melodies that when played together exhibit a serendipitous congruence often enough to be surprisingly pleasant.
The thing is, a mashup is always countermelody, but countermelodies are not mashups.
music.stackexchange.com/users/21900/dwoz
It may be better to use the term polyphony which could include counterpoint and fugues.
In music, polyphony is a texture consisting of two or more simultaneous lines of independent melody, as opposed to a musical texture with just one voice which is called monophony, and in difference from musical texture with one dominant melodic voice accompanied by chords which is called homophony.
SOURCE
music.stackexchange.com/users/7306/Neil Meyer
This can be a Mashup.
A mashup is a song or composition created by blending two or more pre-recorded songs
music.stackexchange.com/users/13492/Zato
Is there any name I can use for two melodies that are being played at the same time?  (The rhythmic part isn't identical) .  For Instance, there is Fables of Faubus by Charles Mingus which starts off with a melody:     and then on 0.15 starts another melody that is being played on top of the other.  How is this named?
Is there any name for two melodies that are being played at the same time?
music.stackexchange.com
<terminology>
music.stackexchange.com/questions/23706
music.stackexchange.com/users/7356/Shevliaskovic
The term counterpoint is used when the melodies that are sounding simultaneously are independent and are more or less of equal importance. Quodlibet  ( Latin: “what you will”) musical composition in which several well-known melodies are combined, either simultaneously or, less frequently, sequentially, for humorous effect.  Mingus himself called this structure of melodies stacked on top of each other - in many of his pieces there are more than two, e.g. in Wednesday Night Prayer Meeting - a "pyramid".  Accompaniment and Mashup is other terms used. There are a number of different terms that may describe this concept, including "counterpoint", "quodlibet", "accompaniment" or a "mashup".
There are a number of different terms that may describe this concept, including "counterpoint", "quodlibet", "accompaniment" or a "mashup".