All articles I found discussing parallel helices are purely speculative with regards to biological significance; but, they are still interesting. Here are some that I found, in addition to the other answer.
Safaee N, Noronha AM, Rodionov D, Kozlov G, Wilds CJ, Sheldrick GM, Gehring K. 2013. Structure of the Parallel Duplex of Poly(A) RNA: Evaluation of a 50 Year-Old Prediction. Angew Chem Into Ed 52:10370-10373.
This paper presents the solved crystal structure of parallel poly(A) RNA and shows that Poly(A) Binding Protein (PABP) promotes parallel duplex formation. A biological role is hypothesized:
As the great majority of eukaryotic messenger RNAs (mRNA) are tagged with 100 to 250 adenines at their 3′ end, the polymorphism of poly(rA) is also relevant for present day cellular processes involving mRNA translation, storage, and decay. Under conditions of cell stress, cellular mRNAs are transported into RNA granules, increasing the local concentration of poly(rA). It is possible that nature evolved proteins such as PABP in part to regulate the occurrence of poly(rA) duplexes in cells.
There are several reviews that discuss the possible role of parallel RNA in the RNA world which revolve around the problem of replication using complementary, anti-parallel strands:
Taylor WR. 2005. Stirring the primordial soup. Nature 434:705.
Some mechanisms for replication in the RNA world have been put forward, and following the current systems of protein polynucleotide synthesis, all involve the creation of a complementary daughter strand using Watson–Crick base-pairing. But from a mechanistic viewpoint, such a model contains a fundamental problem: if a ribopolymerase were to make a complementary copy of itself, it would need to recopy this to obtain a new functional ribopolymerase. This implies that both the ribopolymerase sequence and its complement would have to coexist. But if these two copies came together, the result would be a double stranded Watson–Crick helix (as found in some RNA viruses) — not a new ribopolymerase. Even if both sequences had well determined secondary structures, the perfect complementarity of the Watson–Crick pairing would act as a sink, leading to a sterile population of double-stranded molecules.
The proposed solution is that early RNA polymerases may have created parallel complements to prevent such inhibition:
Taylor WR. 2006. Transcription and translation in an RNA world. Phil Trans R Soc B 361:1751-1760.
Replication strategies. (a) Replication via a reverse complementary strand leads to (b) a stable double-stranded duplex if the two copies meet. (c) Replication via a parallel complementary strand leads to (d) a relatively unstable double-stranded duplex if the two copies meet.
The propagation of information in a nucleic acid strand from one ‘generation’ to the next using Watson–Crick base pairing logically does not have to involve a reverse complementary strand. Providing that there is complementary base pairing, a parallel complement would also propagate the same information...
...all that need change from the viewpoint of replicase is the direction of its progression along the template. The resulting transcript could only be expected to base pair with the template over a short region before parting, but, faced with the problem of irreversible hybridization, this would be a desirable feature of the model.
This paper mentions some proposed functions of parallel stranded (ps) DNA, with accompanying references, but I can only access one of them:
Other functions have been proposed for ps-DNA in gene expression, recombination, RNA processing (14,18,20), the packing of single-stranded and dimeric viral genomes and the function of reverse gyrase (12).
This is the one reference that I could find from the above paper:
Ramsing NB, Jovin TM. Parallel stranded duplex DNA. Nucleic Acids Res 16:6659-6676.
The possibility that ps-RNA might exist is intriguing in view of the rich
structural and functional repertoire of RNA species in general. Three canonical situations in which ps helices could arise by interactions of wholly or partially homologous strands or looping of single stranded nucleic acid of appropriate sequence are shown in Fig. 12. The topological implications of such structures are of great interest, particularly in relation to the potential roles of ps-DNA and ps-RNA in (nonhomologous) recombination, RNA splicing, stabilization of ribosomal RNA, and other cellular processes. In addition, it can be anticipated that specific ligands, particularly proteins, could intervene in order to stabilize and exploit the parallel stranded conformation.