I've been lurking in the parenting forum and hoping someone would address this. I'm a young parent and perhaps not the most qualified to answer, since there are multiple stages of parenthood that I've yet to reach. However, no one else has, so I'll give it a shot. In my opinion being an introverted parent does require extra effort. That being said, parenthood is no less rewarding for introverts. While it can be more difficult, especially if you have an extroverted child, an introvert has a lot to offer as parent.
I'm a stay-at-home-mom. I've just begun educating my five year old at home, and my other child is not yet two years old. I'm absolutely crazy about my children, and I try to put a tremendous effort into being a good mother. I only recently realized that I'm an introvert, but it explains why being a mother has been so challenging at times. Frankly, I thought there was something wrong with me. I couldn't understand how I could love my children so much and yet crave time alone so strongly. I tagged it as selfishness, resolved to work on it, dealt with it as best I could, and went on.
Then a few months ago I stumbled across Susan Cain via an interview with another blogger that I subscribe to. Since then, I've spent a large portion of what little free time I have reading on the subject of introverts. I can't tell how comforting, how marvelously affirming it is to know that I'm not a lousy mom–I'm an introvert! (It has helped me in multiple other facets of life, but that's all rather off-topic.)
I realize that you are looking for perspective, not actual tips for raising children at this point. However, for future use or others who may be lurking, here's what works for me.
–Develop a routine for your family. When everyone needs you and is talking to you at once, it helps to know that you have their afternoon nap or bedtime coming up. Sounds cruel but it's true.
–Get your spouse on board. My husband has precious little free time because of the hours he works. However, he volunteers to watch the kids early one morning each week. I leave, get something to eat, and read, usually. Again, knowing that I'll get some time to myself helps.
–Pick the brains of those around you that have similar personalities. They've been a valuable resource for me. I'm heavily involved in a small church community, so that's one of my main resources.
–If you're struggling with a specific problem, Google it. While I've yet to discover a site dedicated to serving and offering resources to introverted parents, there are many, many articles and blog entries that address the issue. Parenthood rarely takes you where others haven't already been. Fortunately for me, some of those ahead of me are willing to put what they've learned online.
In closing, being an introvert doesn't necessarily mean that you shouldn't have kids. If you're prone to carefully reasoning your way through problems, thinking before you speak, and not following the crowd, you could be fabulous father-material. In my case, it just requires extra planning. Parenthood has its bumps, no matter how you're wired.
Again, I'm sure there are others who can weigh in on this more effectively than I have. I'd love to hear from them.