Hi Elliebirds Mom,
I have an Associate Degree in Early Childhood Education and I also do Parent Education, and I can assure you that your daughter is perfectly fine, really! Her behavior is on par for he age and development.
At 20 months, it is still quite normal for children to do some parallel play, while starting to spread their wings and trying group play. It's also common at this age to be very outgoing at home with mom and dad (and siblings and grandparents, if there are any) because they feel comfortable there. Some kids, while outgoing at home, may take a little longer to warm up with others when away from their comfort zone. There is nothing wrong with this, I promise. Your daughter, over time, may become more outgoing when she is with others, or she may not. That's ok.
I'm willing to bet it's just her personality. I don't think it warrants intervention at tis time, I really don't. I firmly believe it's just a developmental stage, and as she gets older and has more time with other kids, she'll become more comfortable with them.
Do I think you should have socialized her more when she younger? No. That's what you are doing now, and it's a great time to do it. You have done absolutely nothing wrong, so please stop blaming yourself. Just know you are doing the best for your daughter, and I firmly believe you are.
You may want to consider preschool when she turns 3, two or three days a week, or at thievery least, preschool when she turns 4 to help her get ready for school. I'm a firm believer in preschool.
As for what you can do, I have a few ideas. First, keep getting her out with others as much as you can, and sounds like you are doing that. Good for you.
Second, when she is at the park, or mommy and me, and is hesitant about interacting, you can narrate what you see going on around you. For example, You can say "I see Tommy on the swings, that looks like fun. Oh look, Jenny just went down the slide. And Katie is in the sandbox building a castle. Do any of the look lie fun to you?" If you get a response great, if not that's fine too. Just talk about what you see and here around you. This might encourage her to play. But don't force her to do something she isn't ready for, be willing to let her set the pace. Remember, she is still in the parallel play stage.
I have two books to recommend. Both are written by T. Berry Brazelton (one of my favorite go to guys on child development)and Joshua Sparrow M.D.
* Touchpoints; Birth to Three
*Touchpoints: Three to Six
These books have great information on child development milestones and are easy to read. They are great resources and might be helpful for you. I would bet you can get the at the library or through Amazon.
I know this is long, but I wanted to be of some help. Again, please remember, You have done nothing wrong, and you are a great mom!
Good luck!