who designed this park?
The first coherent landscaping was done by Charles Bridgeman.
Hyde Park, London
The first coherent landscaping in Hyde Park began in 1726. It was undertaken by Charles Bridgeman for King George I, but following the king's death the following year, it continued with approval of his daughter-in-law, Queen Caroline. 
when did hyde park open to the public In 1637.
wiki:19523245
Hyde Park, London [SEP] 18th century
Edward Thurlow, 1st Baron Thurlow fought Andrew Stuart in a Hyde Park duel in 1770. Military executions were common in Hyde Park at this time; John Rocque's Map of London, 1746 marks a point inside the park, close to the Tyburn gallows, as "where soldiers are shot." The first coherent landscaping in Hyde Park began in 1726. It was undertaken by Charles Bridgeman for King George I, but following the king's death the following year, it continued with approval of his daughter-in-law, Queen Caroline. Work was undertaken under the supervision of Charles Withers, the Surveyor-General of Woods and Forests. The principal effect of the work was to sub-divide Hyde Park and create Kensington Gardens.