No, there's no real good reason to watch The Originals in any particular order to understand The Vampire Diaries, or vice versa. Unlike other shared universes (say, the Arrowverse), the shows are pretty compartmentalized.
Obviously, The Originals is the spin-off series, so you would definitely want to watch the first few seasons of The Vampire Diaries in order to figure out who the main characters are and what they're up to prior to Klaus heading over to New Orleans.
- Seasons Two introduces Elijah and Klaus, explains a bit about the Original family, and why Klaus is special
- Season Three introduces the rest of the Original family, including Mikael, and explains what happen to Kol and Finn.
- Season Four introduces Hayley, and also includes the "backdoor pilot" for The Originals.
After that, The Originals remains largely separate from The Vampire Diaries. Season Five of the The Vampire Diaries also includes Kol as a recurring character, somewhat hinting at his future in The Originals, but nothing significant.
There are a handful of crossover episodes, but they don't have much of an impact on each other's storylines:
- Tyler Lockwood shows up in New Orleans for two episodes, an event that is hinted at in The Vampire Diaries earlier in the season, but other than knowing that he dumped Caroline there's not much overlap.
- Klaus and Rebekah return to Mystic Falls for the 100th episode of The Vampire Diaries but it's mostly to wrap up their respective storylines with Matt and Caroline -- nothing about New Orleans comes up.
- Stefan visits New Orleans in Seasons 7/3 in a two-parter that's probably the most significant story crossover -- he's running from a vampire hunter -- but if you saw them out of context each episode is pretty understandable on its own.
Beyond that, there aren't even all that many callouts to the other shows. In fact, trying to merge the two continuities together often results in some nonsensical conclusions (neither show tends to even acknowledge the major changes in mythology that pop up every so often).
You can pretty much watch them in whatever order you want and you won't miss anything.