Creating LLC parent company, for existing sole proprietorship
You are thinking in a generally correct direction, except there is no need for parent company complication. This is how it works:
1. Currently your business is you - that's the meaning of "sole proprietorship". The entity is not separate from the owner, meaning all the contracts are done essentially with you, you own the tools, and you carry the liability with everything you own.
2. Once you register an LLC, this LLC is a separate entity from liability point of view, however from tax point of view it's a "disregarded entity", meaning you don't file separate tax return for it, instead you file it together with your personal tax return. This way you enjoy from both the lack of complexity of having additional tax reporting, and at the same time protect your personal assets from business-related liability.
Just remember - once your LLC is formed you need to "transfer" all your business affairs to this LLC, which means all the contracts now must be signed between your clients and the LLC, all the business related assets and property need to be legally transferred to the LLC, and you need to open a separate bank account. Having an Operating Agreement is also an important part of "separating" your LLC from yourself as the owner.