Stick to the 5ths, which shouldn't resolve just to stay in the same 'key' (with parallels you're in the realm of musical pragmatics: very much as case of a) whatever works & b) exceptions proving rules). This was the basis of Medieval Organum, and it's easier to see how that works using an instrument with parallel melody strings - such as a hurdy gurdy, which is conventionally tuned in octaves or unison, but if you tune the strings to 5ths, 4ths, 3rds or minor 3rds the effect is quite stunning (though I once had a discussion with Mary Remnant about this and she regarded such 'abuses' as being 'unmusical' - I suspect she isn't alone in this!). The medieval two-man hurdy gurdy is called the Organistrum, and whilst opinion is divided over how these were actually played, one contemporary illustration clearly shows that all three strings were stopped in tonic / 5th / 8ve parallel harmony with no drones. Other iconographic evidence would indicate at least one drone, though modern reconstructions (there's some especialy ghastly Renn Fair efforts on YouTube) usually have two drones & one melody string which (I feel) defeats the point rather. The Karadeniz Kemence / Pontic Lyra tradition is based around the use of parallel 4ths, though that might sound a little odd to Western ears, at first anyway, though I find this lends itself perfectly to English Folk Song.

On a standard keyboard instrument such parallel playing is never quite as 'organic' somehow, though on the melodica I find intuitive magadizing works a treat, likewise 3rds / 4ths / 5th / 6ths / 7ths. The main problem with melodicas (and why I no longer use them quite as passionately as I did when I first got into them back in 1993) is that they very quickly go out of tune, so you lose those nicely tempered intervals which at first are so seductive.

Maybe it's time to have a go at the fabled $20 Hurdy Gurdy?