Use put-call parity to prove that if a call and a put on the same
stock and having the same expiration date are at-the-money, the call
must be worth more than the put as long as there is time left before
expiration.  Can you provide an intuitive explanation for this result
that does not refer to your proof?  That is, why is it true that a
call is worth more than its twin put if both are at the money?

Because a the execution of a put requires you to purchase and sell
(transaction costs) the underlying value. A call involves just a
single purchase.

Execution - Um no.  The price of the call is higher because there is
an expected return for the stock and that expected return is positive.
 As the stock price is expected to rise over time the options must be
priced to include that rise.  Note that dividends may disrupt this
pricing relationship.

The equation for Put Call Parity is
C + PV(X) = P + S
where
 C = the current market value of the call;
 PV(x) = the present value of the strike price x, discounted from the
expiration date at a suitable risk free rate;
 P = the current market value of the put;
 S= the current market value of the share

It is derived purely using arbitrage arguments. It applies only to
European options, since a possibility of early exercise could cause a
divergence in the present values of the two portfolios. With regard to
your example at the money means X=S

Thus C=PV(S) = P+S

As PV(S)<S (we divide by 1+r typically with r the suitable risk free rate)

C-P = S-PV(S) as said above S-PV(S) is positive thus 
C-P must be positive and then C>P.

This of course only hold if there are time to expiration left,
otherwise S=PV(S) and C-P =0 and in this case C=P, which if at the
money will mean C=P=0. But this is a special rare case.

Intuitively this hold as well ? keep in mind the value of an option is
linked to the probability or chance that it will end in the money and
by how much this value is (X-S for a put and S-X for a call). For a
put this value can be max X, thus if share falls to $0, unlikely but
then we did have a dot com boom! On the call side the value is maximum
? S-X but the share p[rice can go in theory to infinity. For example
warren Buffets Berkshire Hathaway was listed I think for $18 some 35y
ago. But now is worth some $90?000. Thus a 35y call strike $18 would
now be worth ($90 000 -$18) = $89?982. While if it totally blew up and
went bust to zero the put will only be worth $18-0 = $18. The chance
that the call gives you to make insane amounts of money while the
profit of a put is capped, implies that a call must be worth more than
a put everything else being equal.