Is there a word for repeating the starting syllables of a word before completing it, in song?
In Coldplay's song Paradise, for example, the chorus goes "Para-para-paradise, para-para-paradise."
Is there a term for this?
There is a term that would apply, but it does not specifically refer to this as a literary device/singing device, and it doesn't specify that the repetition is at the start of the word.
In linguistics, repeating part of a word is called "partial reduplication". In some languages, this kind of repetition occurs as part of a grammatical process (e.g. in the formation of plural forms of nouns). I found a Lingua Obscura article that uses the same term to refer to this kind of repetition in the context of an English song:
as a creative linguistic process, it won’t surprise you to know that reduplication takes other forms in English, not just contrastive reduplication, as Gomeshi et al. show. For example there’s baby talk or copy reduplication (“choo-choo“), multiple partial reduplication (“hap-hap-happy” as in some song lyrics), [...]
("The Nitty-Gritty on Reduplication: So Good, You Have to Say it Twice," by Chi Luu, October 26, 2016)
This terminology and the example are taken from the paper "Contrastive Focus Reduplication in English (The Salad-Salad Paper)," by Jila Ghomeshi, Ray Jackendoff, Nicole Rosen and Kevin Russell, 2004 (page 309).
The term "multiple partial reduplication" is also used in the paper "Reduplicative Construction in Spanish and English: Pedagogical Grammar Approach," by Maria Teresa Barnes Wales, 2016, which mentions that it is also used in Spanish songs (4.2.3, page 24).
This term doesn't specify the location of the reduplication. Maria Teresa Barnes Wales's paper mentions that reduplication for rhetorical effect in singing often occurs at the end of the word in Spanish, giving the examples "Parchis chis chis" (Parchis, 1979) and "Ella se arrebata bata bata" (Latin Fresh, 1997) (page 24). The word-final variant of multiple partial reduplication can also be found in English songs.