Are there functional examples of parallel DNA double helices?
Surprisingly, a parallel DNA duplex has been reported! In a paper, Tchurikov et al have reported the presence of parallel complementary DNA in the non-coding region of alcohol dehydrogenase gene as well as between two Drosophila DNA sequences. The region, which is ~40 bp long, has 76% bases in same polarity along with complementarity. However, its presence in vivo and its significance are not known (they observed its existwnce in vitro).

Tchurikov et al, in another paper, have reported that parallel complementary RNA in E. coli plays some role in RNA interference and is indeed more effective than antisense RNA in silencing mRNA for gene expression regulation. They also propose the presence of such a system in vivo in E. coli cells. (Seemingly, this paper alone is enough to answer your question since it fulfills all your criteria). 

In another paper, Szabat et al have shown that DNA, 2'-O-MeRNA and RNA oligonucleotides can adopt parallel duplex configuration at pH 5 and lower. Also, presence of LNA stabilizes parallel duplex configuration. This might seem helpful in processes such as RNA interference, though this study too was in vitro (obviously, in vivo LNA is not known).

Many such papers, like Westhof et al, Mohammadi et al, etc. have reported the presence of parallel duplex DNA.

References:

 1. Tchurikov NA, Chernov BK, Golova YB, Nechipurenko YD. Parallel DNA: generation of a duplex between two Drosophila sequences in vitro. FEBS letters. 1989;257(2):415–8. pmid:2479581

 2. Tchurikov, N. A., L. G. Chistyakova, G. B. Zavilgelsky, I. V. Manukhov, B. K. Chernov, and Y. B. Golova. 2000. Gene-specific silencing by expression of parallel complementary RNA in Escherichia coli. J. Biol. Chem. 275:26523–26529

 3. Szabat M, Pedzinski T, Czapik T, Kierzek E, Kierzek R (2015) Structural Aspects of the Antiparallel and Parallel Duplexes Formed by DNA, 2’-O-Methyl RNA and RNA Oligonucleotides. PLoS ONE 10(11): e0143354. doi:10.1371/journal.pone.0143354

 4. Westhof, E., and M. Sundaralingam. 1980. X-ray structure of a cytidylyl-3′,5′-adenosine-proflavine complex: a self-paired parallel-chain double helical dimer with an intercalated acridine dye. Proc. Natl. Acad. Sci. USA. 77:1852–1856.

 5. Mohammadi, S., R. Klement, A. K. Shchyolkina, J. Liquier, T. M. Jovin, and E. Taillandier. 1998. FTIR and UV spectroscopy of parallel-stranded DNAs with mixed A-T/G-C sequences and their A-T/I-C analogues. Biochemistry. 37:16529–16537.