There really shouldn't be a stigma around "mental health", or around any kind of mental illness either. Many people confuse mental health with mental illness, but aside from that, MCI isn't really either of those things.
We all have issues surrounding mental health at various times in our lives - stress from outside influences, for example - but that does not mean we have a mental illness such as bipolar disorder caused by chemical imbalances within the brain. MCI and Altzheimers are diseases which cause damage to the brain, and so really these are different still.
That all said, all are unpleasant, the effects of all of these on a person are quite similar, and it wont be easy news for your parent to hear if you are correct.
You do need to broach the subject soon, because help may be available if he/she does have a problem. I believe the best approach would be to raise your concern directly, reassure them of your support, and try to dispel any "stigma" that they are concerned about.
You could perhaps say:
I've noticed some changes in you recently. You may not be aware of them yourself, but I'm concerned that they could be signs of a cognitive disorder. If these things are caught early they can be helped. Would you let me make an appointment for you to get checked out to make sure you are okay?
I imagine they will want to know more than that. If you go on to say what the "changes" you have noticed are, I would avoid mentioning the "irritability" and other similar symptoms and instead focus on the inarticulate speech. This is because your parent may argue they are not symptoms of a condition. The speech is really the one symptom that is not contestable and least normal.
If they refuse and you believe this is because of "pride" or their ego, perhaps reason this way:
I know this isn't easy to hear and you probably feel that getting a diagnosis would be undignified, but if you do have a cognitive disorder and you don't get diagnosed, you could deteriorate into a much less dignified condition.
If they do feel there is some kind of "stigma" and don't understand the difference between what you think they may have and other mental illnesses, you need to reason that this is something happening to them. Some of the older generations don't believe in mental illnesses, or believe that they can be overcome with willpower. You need your parent to understand that this is not a "weakness" of their mind, but rather a disease that is attacking their mind, and could possibly be stopped if they get help.