Parameterized report links designed for Classic do not work in Lightning, and those designed for Lightning do not work in Classic.
There's a solution for hyperlink formula fields, but it doesn't work for Custom Links. To make a hyperlink formula field work in both environments, you embed a formula expression that looks at the
$User.UIThemeDisplayed context variable, which is "Theme4d" on Lightning. The formula can then conditionally render a link that works for Lightning or for Classic. (Here's a fuller explication, not by me).
Unfortunately, applying this solution to Custom Links just plain doesn't work - fields and text that are merged into a Custom Link from a formula expression are URL-escaped, which destroys the link if you try something like
"/lightning/r/Report/MY_ID/view?fv0=" & Account.Id ,
"/MY_ID?pv0="& Account.Id ) }
This just yields
You have sent us an Illegal URL or an improperly formatted request.
(Note the URL escaping).
I've tried a number of routes around this but haven't found anything that actually works.
The delivered Idea for parameterized report links in Lightning suggests that this situation will probably not be fixed:
Current solution in Classic is not changing, but it was something we never officially supported.