In terms of estimation, both the Cox model and parametric models that belong to the proportional hazards (PH) family will not be affected by a high proportion of censored observations.
The reason is that the likelihood (or partial likelihood for Cox) is a cumulative product where only individuals who experience the event provide a contribution. Censored observations only provide a passive contribution in the definition of the individuals at risk for each event time.
In terms of interpretation, you will have to specify that the HR you are presenting summarizes a given comparison within a specific window of time where only p% of events were observed. In practice, let's say that in one year you have more events and redo the analyses, the HR will likely change (even under PH being valid).