Do the parents of my puppy need to have show titles?
The long answer, believe it or not, is no. But please, read on.
The short answer...when in doubt, YES. It doesn't matter that you won't be showing your puppy. What matters here is that if you are not sure about meeting the parents and judging for yourself if the parents adhere to the registered standard for your breed, than show titles do that job for you. During a show, a judge measures the dog - your future puppy momma or pappa - against the accepted standard and determines they have no disqualifications. This is important for each and every breed.
Now, the long answer. Show titles do not mean the parents are healthy. Show titles do not mean the parents will produce puppies that look like them - meaning puppies that also adhere to the standard without disqualifications. Show titles do not mean the parents have good temperaments.
More important, to maintaining any breed, than show titles are the health, the production of the genetics, and the temperaments or drives associated with the breed. Before showing dogs was a popular activity, the purity of dog breeds was maintained by selectively breeding those dogs who looked the way the breed was intended to look and performed the way the breed was intended to perform without fault.
Let's say -- You're looking for a Rottweiler puppy. You find a highly titled and revered kennel with puppies due soon and you take one home. Those parents had those show titles! But, the father soon dies at a very young age of 4 from congenital heart failure and the mother is dysplasitc and crippled before she is 6. How valuable are those show titles now? Not so much. And, sadly, these parents are not contributing anything valuable to the breed of Rottweilers but instead are repeating unhealthy conditions in their future generations.
Perhaps the health is fine, but gosh...that puppy doesn't look a thing like the parents. The muzzle is long and pointed, the puppy grows into a very tall dog, the color is way too light, and your puppy has a big pointed knot on the top of her forehead instead of the well defined square skull you expected. These parents did not reproduce their appearance at all in their offspring - assuming both that well titled mother and father had none of these faults. Instead, the stronger recessive traits in the genetics of these parents produced something with faults that are disqualifications for the breed standard and probably not at all what you had envisioned when you went searching for a Rottweiler puppy.
Finally, what about that temperament? Perhaps the breeder had a reason for why you could not meet his prized, titled, soon to be parents and you went along with that. Perhaps it turns out the mother is so aggressive she must be artificially bred because she will attack any other dog who goes near her - whether she is in season or not. Or perhaps the male is so people aggressive it would be a liability to allow future potential puppy families to meet him. And so you take your puppy home and end up with something that, in-spite of your best socializing efforts, is dog or people aggressive. While most behavioral issues can be trained or socialized away from, the injustice is breeding parents whose temperament is not at all acceptable within the breed and repeating that proximity for aggression in future generations.
Titles do matter. They tell us a lot. I myself want to know the titles, when they were won, at what age, under which judge, and the location. I can learn a lot from those details.
But more than titles, I can learn a lot from documented health clearances and certifications, seeing multiple generations of parents, grandparents, nieces, nephews, siblings and so on related to the soon-to-be parents of the upcoming litter in question, and by meeting those dogs myself in person and feeling comfortable with them near me and near other dogs showing us the biddable and confident temperament associated with a stoic breed.