We do Salesforce API requests in behalf of our clients. Our clients continue doing API requests independently from us.
We're looking for a way that our API calls doesn't count against the concurrent API call limits.
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Every API call to salesforce will count towards your limit. You can consider upgrading to a higher-tier edition of salesforce or buying more licenses in order to increase your limit but there's always a limit.
You'll likely want to consider a different integration architecture altogether if you wish to reduce your calls somehow. Some ideas off the top of my head: