More power to national parliaments is a democracy–enhancing move only if the national decision-making occurs in areas that have little impact on the others. If the opposite is true, it diminishes the control we have over the important decisions.
David Cameroon has announced his requests to the EU. One of them is a larger role for national parliaments in shaping EU regulations. Although national parliaments may be the most democratic institutions we have, the areas where such a "repatriation of powers" occurs must be chosen carefully. Otherwise it may weaken the democracy instead. There are two main reasons.
First, and most importantly, only some decisions can be made by national parliaments. Many decisions can not, such as international treaties, trade agreements and international law. Almost by definition this type of legislation is not made by individual states. Analogously, parliaments do not vote on what the other countries do, be it introduction of incompatible regulations, or conducting irresponsible fiscal policy in the hope that we will bail them out later. Neither have we much room to vote about technical standards, such as Apple's new operating system or airport security procedures.
A number of major problems EU currently faces cannot be solved at national level. For instance, the refugee inflow must either be received, or stopped, at the EU border. The current country-based decision making essentially allows Hungarian parliament to decide over number of refugees in Serbia and Croatia, and, if German "national parliament" decides to close it's borders, it has severe impact on Greece and Italy. Analogously, the "single market", one of the pillars of EU, is a form of extended free-trade agreement, a large number of common standards for product labeling, food safety, and labor treatment. By definition, agreement is something we cannot do alone.
More powerful national parliaments in this type of decisions will weaken EU without strengthening the member–states. We will notice more small inconveniences, such that you cannot use certain mobile services in another EU country, or your business has to hassle with incompatible regulations across the border. Negotiations among 28 sovereign states are far more slow and costly than central decisionmaking, and there is little incentive to overcome even small special interests in the name of a common good. National politicians are elected to stand for "national interests", and all 28 national interests are seldom aligned with the common one. The underlying problem is interdependency, our decisions may influence others even if the others do not belong to our "nation". Such decisions should be done by inclusive higher level bodies, such as European Parliament.
Enfranchisement of EU Migrants
The other, and currently less important reason, is enfranchisement of EU migrants. This is a large group of people who cannot vote in national elections but is subject to the corresponding national legislation. This is simply not fair. I think the first-best solution would be to give EU migrants voting rights rather soon after moving to another country (say, in 2 years). But until it happens, decision–making by EU institutions offers this group more say about their lives.
Finally, even EU is too small for many important decisions. Many contemporary problems, such as global trade agreements, climate change, or the puzzle of Middle–East, cannot be solved at EU level either. We need global governance more than ever before. How to achieve this in a democratic way is one of the big challenges of our time.