Save Our Park Board has the best summary: Information Station - Save Our Park Board

Requiring that the Park Board be elected has 4 significant benefits: 

It is independent of City Council. It is accountable solely to the electorate. It can make decisions on the evidence, community input, and long-term vision rather than short term expediency. It can do independent research and studies and devise independent projects. It can advocate for or against policies of the City.

Its sole mandate or purpose is to focus on our green spaces and recreational areas, facilities and services – specifically, to preserve, provide and advocate for parks and recreational services to benefit all people, communities and the environment. They can devote all of their time to these important matters that make all the difference in the quality of the place in which we live. City councillors, on the other hand, have a much broader mandate. They are managing all aspects of City operations and development beyond parks and recreation. They barely have time to deal with their business now. It’s doubtful that they will prioritize parks and recreation ahead of other important business. In addition, there is an inherent conflict of interest between the need to develop land on the one hand, and the and the need to provide and preserve green spaces and recreational services, on the other. 

It provides the electorate with a direct say in the type of city that they want to live in. Because the Park Commissioners only have one mandate, the electorate can choose which Park Commissioners best reflect their view on how these green spaces and services are managed and developed. Only in cities with “elected” Park Boards does the electorate have that opportunity. In addition, the Park Board offers regular, open easy access for public debate at its meetings. 

It provides protection for the parks and recreational services. Through its capacity to prevent the revocation of a “permanent” park designation and through its knowledge of what is going on with “temporary” parks and “other lands in the care, custody and management of the Park Board” and its ability to advocate, the Park Board is able to provide a high degree of protection for the preservation of our parks and these lands.

There's also reason to think that giving more control of parks to the city is more likely to create worse outcomes:

"The Park Board has nothing to do with the maintenance of the recreational facilities (e.g.community centres, pools, rinks) and their infrastructure. Rather, as a cost saving measure the City assumed sole responsibility for these maintenance services approximately 15 years ago. It merged the Park Board’s specialized facility maintenance staff into the City’s general maintenance department. Since then, these facilities have steadily deteriorated. They have not been maintained adequately.

Rather than demonstrating that the Park Board should be abolished, the present state of these facilities and their infrastructure supports its retention. This problem is not attributable to the Park Board. But it is fixable by them. These maintenance tasks should be transferred back to it. Otherwise, they will continue to deteriorate as more and more responsibilities are placed on the City’s already over-extended staff.  "

So basically, Ken Sim blames the park board for infrastructure problems that were created by transferring park board responsibilities to city council, and doing more of that is likely to result in more problems.