Bhutan will have pleasant weather in June. No, a lot of trekking is not required to visit the main attractions. You can skip the Tigers nest monastery if your elderly parents cannot hike. It is quite a hike and might not be ideal for elders. Apart from that, there are dzongs which do not need a lot of trekking. You can read more about Bhutan on
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You dont have to trek at all to visit 80% of the more famous tourist spots in Bhutan. Weather will be comfortable, with chances of rain.
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Bhutan is a warm place during may-june. Bhutan is famous mostly for trekking places. yes almost attractive places in Bhutan needs trekking. well enjoy your trip