First, mixed languages, pidgins, and creoles are all different things, so you should beware using the term "mixed language" as a synonym for pidgin or creole. The overarching term that would include all three of these terms and others is contact language.
Now, the following from Baker and Mous (1994, p. 5) attempts to define a mixed language:
"A very rough approximation is that a mixed language has its lexicon and grammar from different sources. On the basis of the lexicon one would classify such languages as belonging to one language family and on the basis of morphology, syntax, and general grammatical characteristics one would classify them as belonging to another language family."
The above shows that mixed languages are a melange, but there is reference to parent languages, and it does not include those things which separate a creole from a mixed language. One such difference, according to Mark Seeba, is that mixed languages, unlike creoles, tend to retain the morphological complexity of their input languages (1997).
From here on I discuss the main question at hand: do contact languages have families? The short answer is that it depends. Research can be used to say yes or no. The long answer follows below.
I suppose that daughter languages are typically shown as being born from one parent language, as we might find with English stemming from a Germanic lineage, but even this seems overly simplistic when we consider arguments like Bailey and Maroldt's (1977) Middle English creole hypothesis, which argues that Middle English was a creole born from language contact (primarily Old English and Norman French, but with evident tertiary contact considered key to pidginisation/creolisation).
Because creoles tend to be the result of contact between at least three languages (although we often only refer to the superstrate and substrate), the act of genealogical tracing becomes more difficult but not impossible or unheard of. For example, I have personally seen language genealogy trees linking Gullah and Black English to a parent African language (i.e., West African Pidgin English) based on various lexical and syntactic rules. I cannot cite the direct source at the moment, but I have seen it used in at least one book in actual tree form, derived into the same structure in my Sociolinguistics seminar, and often explained in expanded form by linguists like John Rickford, Russel Rickford, and Lisa Green. I have read similar accounts for Louisiana French Creole by Thomas Klingler and others.
One could posit a very strong argument for the parentage of any given creole simply through research and writing. Personally, I am on the side that would consider a creole to be the daughter of multiple languages with roots tied to both lineages.
For more information and a wonderful, easy read about pidgins, creoles, and other contact languages, refer to Mark Sebba's, "Contact Languages: Pidgins and Creoles."
